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3 Without a calculator, large calculations, multiplication, division, squares, and roots can be done using a log table. Common logarithms (base 10) show the exponent needed to reach a number. Log tables help find values like log 125 approximately.

4 <h2>What is the Log Table?</h2>

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6 ▶

7 A logarithm table is a mathematical reference used to determine the logarithm of a<a>number</a>with respect to a specific<a>base</a>. Separate<a>log</a><a>tables</a>exist for different bases such as 10, e (Euler’s number), and 2. Among these, the base-10 logarithm called the common logarithm is the most frequently used and is denoted by log or log₁₀.

8 A standard common logarithm (base-10) table is arranged into<a>multiple</a>columns for easy reference:

9 <ul><li>The main column contains two-digit numbers from 10 to 99, representing the first two digits of the number. </li>

10 <li>The next column displays logarithmic values corresponding to the third digit (0-9). </li>

11 <li>The<a>mean</a>difference column provides<a>minor</a>corrections to the fourth digit to improve accuracy.</li>

12 </ul><h2>Difference Between Log and Antilog Table</h2>

13 Log and antilog tables are mathematical reference tools used to simplify complex calculations. The difference between the log and antilog tables is:

14 Log TableAntilog TableA log table helps convert numbers into their logarithmic form. An<a>antilog table</a>performs the inverse operation, converting logarithmic values back into numbers. A logarithm table is typically based on either base 10 or base e. An antilogarithm table is typically based on either base 10 or base e. Contains logarithmic values of numbers. Contains antilogarithmic (exponential) values of logarithms. Helpful in simplifying<a>multiplication</a>and<a>division</a>. Helpful in solving exponentiation and<a>power</a>-related calculations. Example: log10(2) = 0.3010 Example: antilog10(0.3010) = 2<h2>How to Use a Log Table?</h2>

15 To use a logarithm table (base 10) to calculate the logarithm of a number, we must understand that a logarithm consists of two components: characteristic and mantissa, and both are separated by a<a>decimal</a>point.

16 For instance, log10 23.78 = 1.3762, which can be found by using a scientific<a>calculator</a>or a common logarithmic table (base 10). In this value, 1 is the characteristic (the<a></a><a>integer</a>part) and 0.3762 is the mantissa (the fractional part) of the given number.

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19 <h2>What are the Characteristics of a Logarithm of a Number?</h2>

18 <h2>What are the Characteristics of a Logarithm of a Number?</h2>

20 The characteristic of a logarithm is the<a>whole number</a>part of the logarithm. It represents the<a>exponent</a>of 10 when the number is written in scientific notation. Depending on the number, the characteristic can be positive, zero, or negative.

19 The characteristic of a logarithm is the<a>whole number</a>part of the logarithm. It represents the<a>exponent</a>of 10 when the number is written in scientific notation. Depending on the number, the characteristic can be positive, zero, or negative.

21 The characteristic does not depend on the log table. It can be found easily by counting digits using simple rules:

20 The characteristic does not depend on the log table. It can be found easily by counting digits using simple rules:

22 <ul><li>If the number is<a>greater than</a>1: Characteristic = (number of digits to the left of the decimal) - 1.</li>

21 <ul><li>If the number is<a>greater than</a>1: Characteristic = (number of digits to the left of the decimal) - 1.</li>

23 </ul><ul><li>If the number is<a>less than</a>1:Characteristic = -(number of zeros after the decimal before the first non-zero digit + 1).

22 </ul><ul><li>If the number is<a>less than</a>1:Characteristic = -(number of zeros after the decimal before the first non-zero digit + 1).

24 </li>

23 </li>

25 </ul>For example:

24 </ul>For example:

26 For 23.78:

25 For 23.78:

27 Two digits are to the left of the<a>decimal</a>. Characteristic \(= 2 - 1 = 1.\)

26 Two digits are to the left of the<a>decimal</a>. Characteristic \(= 2 - 1 = 1.\)

28 For 0.172:

27 For 0.172:

29 No zeros after the decimal before the first non-zero digit. Characteristic \(= -(0 + 1) = -1\).

28 No zeros after the decimal before the first non-zero digit. Characteristic \(= -(0 + 1) = -1\).

30 <h2>What is Mantissa (Only Positive)?</h2>

29 <h2>What is Mantissa (Only Positive)?</h2>

31 A log table, which is always a positive number prefixed by a decimal point, can be used to determine the mantissa of the logarithm of a number. Now, let us try to figure out the mantissa of the log10 (0.001724).

30 A log table, which is always a positive number prefixed by a decimal point, can be used to determine the mantissa of the logarithm of a number. Now, let us try to figure out the mantissa of the log10 (0.001724).

32 Step 1:Find the first non-zero digit of the given number. Here, the first non-zero digit is 1.

31 Step 1:Find the first non-zero digit of the given number. Here, the first non-zero digit is 1.

33 Step 2:Ignore the decimal point and take the next four digits from the first non-zero digit. Hence, we have 1724.

32 Step 2:Ignore the decimal point and take the next four digits from the first non-zero digit. Hence, we have 1724.

34 Step 3:The log table’s row number is represented by the first two digits (17) in the number 1724, while the column number is represented by the third digit (2).

33 Step 3:The log table’s row number is represented by the first two digits (17) in the number 1724, while the column number is represented by the third digit (2).

35 Therefore, find the value from the table where this row and column intersect. Here, the<a>matching</a>number is 2355.

34 Therefore, find the value from the table where this row and column intersect. Here, the<a>matching</a>number is 2355.

36 Step 4:If the number has only three digits, the table value is directly the mantissa.

35 Step 4:If the number has only three digits, the table value is directly the mantissa.

37 Example: To find log₁₀(172), look up 172 in the table. The value 0.2355 is the mantissa. No mean difference is needed.

36 Example: To find log₁₀(172), look up 172 in the table. The value 0.2355 is the mantissa. No mean difference is needed.

38 If the number has a fourth digit, use the mean difference from the table corresponding to that digit.

37 If the number has a fourth digit, use the mean difference from the table corresponding to that digit.

39 Example: For 1724, the fourth digit is 4. Find the mean difference for 4 in row 17 and add it to the table value to get the mantissa. Here, 10 is the corresponding mean difference.

38 Example: For 1724, the fourth digit is 4. Find the mean difference for 4 in row 17 and add it to the table value to get the mantissa. Here, 10 is the corresponding mean difference.

40 Step 5:After that,<a>sum</a>the two numbers from step 3 and step 4.

39 Step 5:After that,<a>sum</a>the two numbers from step 3 and step 4.

41 \(2355 + 10 = 2365 \)

40 \(2355 + 10 = 2365 \)

42 Step 6:Place a decimal point in front of the number to get the mantissa.

41 Step 6:Place a decimal point in front of the number to get the mantissa.

43 Hence, the mantissa of log 0.001724 is 0.2365. Since 0.001724 < 1, we calculate the characteristic.

42 Hence, the mantissa of log 0.001724 is 0.2365. Since 0.001724 < 1, we calculate the characteristic.

44 The<a>formula</a>is:

43 The<a>formula</a>is:

45 Characteristic = -(number of zeros after the decimal point before a non-zero digit + 1) \( = -(2 + 1) = -3\)

44 Characteristic = -(number of zeros after the decimal point before a non-zero digit + 1) \( = -(2 + 1) = -3\)

46 Logarithm = characteristic + mantissa \( = -3 + 0.2365 = -2.763\)

45 Logarithm = characteristic + mantissa \( = -3 + 0.2365 = -2.763\)

47 Hence, log10 (0.001724) ≈ -2.763.

46 Hence, log10 (0.001724) ≈ -2.763.

48 Keep in mind that we can only find the mantissa if the number from step 2 has four or fewer digits. If the number is less than 4 digits, like 18, treat it as 1800 and determine the mantissa.

47 Keep in mind that we can only find the mantissa if the number from step 2 has four or fewer digits. If the number is less than 4 digits, like 18, treat it as 1800 and determine the mantissa.

49 <h2>How to Find the Logarithm of a Number by Log Table?</h2>

48 <h2>How to Find the Logarithm of a Number by Log Table?</h2>

50 We must follow certain steps to use a logarithm table to determine a number’s logarithm.

49 We must follow certain steps to use a logarithm table to determine a number’s logarithm.

51 Step 1:Find the characteristic of the number, which is the integer part of the logarithm.

50 Step 1:Find the characteristic of the number, which is the integer part of the logarithm.

52 Step 2:Find the mantissa using a log table.

51 Step 2:Find the mantissa using a log table.

53 Step 3:Add the characteristic and mantissa together.

52 Step 3:Add the characteristic and mantissa together.

54 To understand this, let us look at an example. For instance, the given number is 23.78.

53 To understand this, let us look at an example. For instance, the given number is 23.78.

55 Step 1:We apply the following formula, since the given number is greater than 1:

54 Step 1:We apply the following formula, since the given number is greater than 1:

56 Characteristic = (number of digits to the left of the decimal point - 1) Here, 2 digits precede the decimal point in 23.78. Hence, characteristic \(= 2 - 1 = 1 \)

55 Characteristic = (number of digits to the left of the decimal point - 1) Here, 2 digits precede the decimal point in 23.78. Hence, characteristic \(= 2 - 1 = 1 \)

57 Step 2:Since 23.78 is the given number, ignore the decimal point and take it as 2378. The first two digits = 23 Thus, the log table’s row number is 23.

56 Step 2:Since 23.78 is the given number, ignore the decimal point and take it as 2378. The first two digits = 23 Thus, the log table’s row number is 23.

58 The third digit = 7 Hence, 7 is the column number. Next, determine where row 23 and column 7 intersect. Therefore, the corresponding number is 3747.

57 The third digit = 7 Hence, 7 is the column number. Next, determine where row 23 and column 7 intersect. Therefore, the corresponding number is 3747.

59 Now, the fourth digit = 8 Find the mean difference for 8 in row 23. Therefore, the mean difference is 15.

58 Now, the fourth digit = 8 Find the mean difference for 8 in row 23. Therefore, the mean difference is 15.

60 Then, add the mean difference of 15 and the number, 3747.

59 Then, add the mean difference of 15 and the number, 3747.

61 \(3747 + 15 = 3762 \)

60 \(3747 + 15 = 3762 \)

62 Place a decimal point for the result. Mantissa = 0.3762

61 Place a decimal point for the result. Mantissa = 0.3762

63 Step 3:To determine the logarithm of 23.78, add the characteristic and mantissa.

62 Step 3:To determine the logarithm of 23.78, add the characteristic and mantissa.

64 \(1 + 0.3762 = 1.3762 \) \(log (23.78) = 1.3762\)

63 \(1 + 0.3762 = 1.3762 \) \(log (23.78) = 1.3762\)

65 <h2>How to Use a Log Table in Calculations?</h2>

64 <h2>How to Use a Log Table in Calculations?</h2>

66 To perform complex calculations like multiplication, division, and exponents, we can use logarithms, especially without a calculator.

65 To perform complex calculations like multiplication, division, and exponents, we can use logarithms, especially without a calculator.

67 The properties of logarithms are as follows:

66 The properties of logarithms are as follows:

68 <ul><li>\(\log(mn) = \log m + \log n\)</li>

67 <ul><li>\(\log(mn) = \log m + \log n\)</li>

69 <li>\( \log\left(\frac{m}{n}\right) = \log m - \log n \)</li>

68 <li>\( \log\left(\frac{m}{n}\right) = \log m - \log n \)</li>

70 <li>\(\log(m^n) = n \log m\) </li>

69 <li>\(\log(m^n) = n \log m\) </li>

71 </ul>Let us now examine how to use a log table in calculations using an example. Find (17.56 × 37) / (4.75 × 24) by applying the logarithm table approach.

70 </ul>Let us now examine how to use a log table in calculations using an example. Find (17.56 × 37) / (4.75 × 24) by applying the logarithm table approach.

72 Step 1:Determine the logarithm of the given<a>expression</a>using the logarithm properties. \( \log\left(\frac{m}{n}\right) = \log m - \log n \)

71 Step 1:Determine the logarithm of the given<a>expression</a>using the logarithm properties. \( \log\left(\frac{m}{n}\right) = \log m - \log n \)

73 \((17.56 × 37) / (4.75 × 24) \)

72 \((17.56 × 37) / (4.75 × 24) \)

74 \(= log (17.56 × 37) - log (4.75 × 24) \)

73 \(= log (17.56 × 37) - log (4.75 × 24) \)

75 \(= log (17.56) + 7 log (3) - (log (4.75) + 4 log (2)) \)

74 \(= log (17.56) + 7 log (3) - (log (4.75) + 4 log (2)) \)

76 \(= log (17.56) + 7 log (3) - log (4.75) - 4 log (2)\)

75 \(= log (17.56) + 7 log (3) - log (4.75) - 4 log (2)\)

77 We can now determine each logarithm’s characteristic and mantissa using the log table.

76 We can now determine each logarithm’s characteristic and mantissa using the log table.

78 x

77 x

79 Characteristic

78 Characteristic

80 Mantissa

79 Mantissa

81 log x (characteristic + mantissa)

80 log x (characteristic + mantissa)

82 17.56

81 17.56

83 1

82 1

84 0.2445 1.2445 3

83 0.2445 1.2445 3

85 0 0.4771 0.47714.75

84 0 0.4771 0.47714.75

86 0 0.6767 0.67672

85 0 0.6767 0.67672

87 0 0.301 0.301Next, add the values:

86 0 0.301 0.301Next, add the values:

88 \( = 1.2445 + 7 (0.4771) - 0.6767 - 4 (0.301)\)

87 \( = 1.2445 + 7 (0.4771) - 0.6767 - 4 (0.301)\)

89 \( = 1.2445 + 3.3397 - 0.6767 - 1.2040\)

88 \( = 1.2445 + 3.3397 - 0.6767 - 1.2040\)

90 \( = 4.5842 - 1.8807 \)

89 \( = 4.5842 - 1.8807 \)

91 \( = 2.7035\)

90 \( = 2.7035\)

92 Step 2:Next, use the anti-log table to determine the above number’s antilogarithm.

91 Step 2:Next, use the anti-log table to determine the above number’s antilogarithm.

93 Antilog (2.7035), which is equal to 102.7035

92 Antilog (2.7035), which is equal to 102.7035

94 \(10^{2.7035} = 102 × 10^{0.7035} \)

93 \(10^{2.7035} = 102 × 10^{0.7035} \)

95 \(= 100 × Antilog (0.7035) \)

94 \(= 100 × Antilog (0.7035) \)

96 Now, estimate Antilog (0.7035):

95 Now, estimate Antilog (0.7035):

97 Antilog (0.7035) 5.057

96 Antilog (0.7035) 5.057

98 So, \(10^{2.7035} ≈ 100 × 5.057 = 505. 7 \)

97 So, \(10^{2.7035} ≈ 100 × 5.057 = 505. 7 \)

99 Thus, (17.56 × 37) / (4.75 × 24) is approximately equal to 505.7

98 Thus, (17.56 × 37) / (4.75 × 24) is approximately equal to 505.7

100 <h2>How to Use a Log Table for Natural Logarithms?</h2>

99 <h2>How to Use a Log Table for Natural Logarithms?</h2>

101 To determine natural logarithms (logarithms with base e, where e ≈ 2.718), we use natural log tables, often labeled as ln tables. These tables list the values of ln(x), where x is a positive<a>real number</a>. Unlike common log tables, natural log tables are specifically designed to help find logarithms to base e.

100 To determine natural logarithms (logarithms with base e, where e ≈ 2.718), we use natural log tables, often labeled as ln tables. These tables list the values of ln(x), where x is a positive<a>real number</a>. Unlike common log tables, natural log tables are specifically designed to help find logarithms to base e.

102 Using a change of base rule, e = 2.718 can be written as follows:

101 Using a change of base rule, e = 2.718 can be written as follows:

103 \( (log x) / (log e) = (log x) / (log 2.718) \)

102 \( (log x) / (log e) = (log x) / (log 2.718) \)

104 Next, use the logarithm table to determine the values of log x and log 2.718 separately, then divide the results.

103 Next, use the logarithm table to determine the values of log x and log 2.718 separately, then divide the results.

105 For example, determine the value of ln 10.

104 For example, determine the value of ln 10.

106 log10 e ≈ 0.4343, this is the common logarithm of Euler’s number

105 log10 e ≈ 0.4343, this is the common logarithm of Euler’s number

107 e ≈ 2.718

106 e ≈ 2.718

108 \(\ln 10 = \frac{\log_{10} x}{\log_{10} e} = \frac{\log_{10} 10}{\log_{10} 2.718}\)

107 \(\ln 10 = \frac{\log_{10} x}{\log_{10} e} = \frac{\log_{10} 10}{\log_{10} 2.718}\)

109 \( = (1.0000) / (0.4343) \)

108 \( = (1.0000) / (0.4343) \)

110 Hence, ln \(10 = 1.0000 / 0.4343 = 2.3026 \)

109 Hence, ln \(10 = 1.0000 / 0.4343 = 2.3026 \)

111 <h2>Logarithmic Table 1 To 10</h2>

110 <h2>Logarithmic Table 1 To 10</h2>

112 A common logarithm table lists logarithms with base 10, also known as decimal logarithms. These values show the power to which 10 must be raised to obtain a given number. The log values for the numbers from 1 to 10 are:

111 A common logarithm table lists logarithms with base 10, also known as decimal logarithms. These values show the power to which 10 must be raised to obtain a given number. The log values for the numbers from 1 to 10 are:

113 Number(x) log10(x) 1 0 2 0.3010 3 0.4771 4 0.6020 5 0.6989 6 0.7781 7 0.8450 8 0.9030 9 0.9542 10 1<h2>Tips and Tricks to Master Log Table</h2>

112 Number(x) log10(x) 1 0 2 0.3010 3 0.4771 4 0.6020 5 0.6989 6 0.7781 7 0.8450 8 0.9030 9 0.9542 10 1<h2>Tips and Tricks to Master Log Table</h2>

114 Log tables can seem challenging for children, but with parental guidance, they become manageable. By understanding basics, practicing step-by-step, and using simple tips, parents can help their child build confidence,<a>accuracy</a>, and a positive attitude toward logarithms.

113 Log tables can seem challenging for children, but with parental guidance, they become manageable. By understanding basics, practicing step-by-step, and using simple tips, parents can help their child build confidence,<a>accuracy</a>, and a positive attitude toward logarithms.

115 <ul><li>Before your child uses a log table, make sure they understand what logarithms represent, that it’s the power to which a number must be raised to get another number. Encourage them to see it as the “reverse” of exponentiation. </li>

114 <ul><li>Before your child uses a log table, make sure they understand what logarithms represent, that it’s the power to which a number must be raised to get another number. Encourage them to see it as the “reverse” of exponentiation. </li>

116 <li>Start with easy numbers like 2, 3, 5, 10 to practice finding logs. Once confident, move to more<a>complex numbers</a>. Early success boosts confidence. </li>

115 <li>Start with easy numbers like 2, 3, 5, 10 to practice finding logs. Once confident, move to more<a>complex numbers</a>. Early success boosts confidence. </li>

117 <li>Make a small reference sheet with steps and tips for using the log table. This boosts independence and reduces errors. </li>

116 <li>Make a small reference sheet with steps and tips for using the log table. This boosts independence and reduces errors. </li>

118 <li>Some numbers have patterns in logs (like<a>powers of 10</a>or multiples of 2). Recognizing patterns helps your child estimate and cross-check answers quickly. </li>

117 <li>Some numbers have patterns in logs (like<a>powers of 10</a>or multiples of 2). Recognizing patterns helps your child estimate and cross-check answers quickly. </li>

119 <li>Parents can help children estimate approximate answers mentally before checking in the log table. This improves number sense and reduces dependency. </li>

118 <li>Parents can help children estimate approximate answers mentally before checking in the log table. This improves number sense and reduces dependency. </li>

120 <li>Teachers can explain that a log x calculator or a logarithm calculator can be used to verify the answer. </li>

119 <li>Teachers can explain that a log x calculator or a logarithm calculator can be used to verify the answer. </li>

121 <li>Parents may provide a simple logarithm book to help students become comfortable reading tables rather than relying solely on electronic devices. </li>

120 <li>Parents may provide a simple logarithm book to help students become comfortable reading tables rather than relying solely on electronic devices. </li>

122 <li>Teachers can provide structured<a>worksheets</a>that focus on reading mantissas, characteristics, and mean differences, reinforcing correct table usage.</li>

121 <li>Teachers can provide structured<a>worksheets</a>that focus on reading mantissas, characteristics, and mean differences, reinforcing correct table usage.</li>

123 </ul><h2>Common Mistakes and How to Avoid Them on Log Table</h2>

122 </ul><h2>Common Mistakes and How to Avoid Them on Log Table</h2>

124 Without a calculator, complex calculations like multiplication, division, and exponents can be solved using logarithm tables. However, students make mistakes when they work with log tables. Here are some common errors and their helpful solutions to avoid confusion and incorrect answers.

123 Without a calculator, complex calculations like multiplication, division, and exponents can be solved using logarithm tables. However, students make mistakes when they work with log tables. Here are some common errors and their helpful solutions to avoid confusion and incorrect answers.

125 <h2>Real-Life Applications of Log Table</h2>

124 <h2>Real-Life Applications of Log Table</h2>

126 Understanding how log tables work will help students apply them in various situations and fields. The practical uses of the log table are listed below:

125 Understanding how log tables work will help students apply them in various situations and fields. The practical uses of the log table are listed below:

127 <ul><li>Science:In science and chemistry, log tables are used to measure acidity or alkalinity, calculate radioactive decay, and solve problems that involve very small or very large quantities. They help scientists and students perform accurate calculations quickly without relying solely on calculators. </li>

126 <ul><li>Science:In science and chemistry, log tables are used to measure acidity or alkalinity, calculate radioactive decay, and solve problems that involve very small or very large quantities. They help scientists and students perform accurate calculations quickly without relying solely on calculators. </li>

128 <li>Engineering:Engineers use log tables to manage calculations involving sound intensity, signal strength, and electrical circuits. Before calculators, log tables were essential for designing machines, amplifiers, and communication systems. They help convert complicated multiplication and division into simpler steps. </li>

127 <li>Engineering:Engineers use log tables to manage calculations involving sound intensity, signal strength, and electrical circuits. Before calculators, log tables were essential for designing machines, amplifiers, and communication systems. They help convert complicated multiplication and division into simpler steps. </li>

129 <li>Space Science:Log tables are important in fields like geology, seismology, and astronomy. They help scientists measure earthquake magnitudes, compare star brightness, and calculate distances in space. Using log tables makes it easier to handle extremely large numbers efficiently. </li>

128 <li>Space Science:Log tables are important in fields like geology, seismology, and astronomy. They help scientists measure earthquake magnitudes, compare star brightness, and calculate distances in space. Using log tables makes it easier to handle extremely large numbers efficiently. </li>

130 <li>Computer Science and Information:Even in computer science, log tables were historically used to analyze algorithms and solve problems involving large datasets. They help programmers understand how numbers grow and how processes scale efficiently. </li>

129 <li>Computer Science and Information:Even in computer science, log tables were historically used to analyze algorithms and solve problems involving large datasets. They help programmers understand how numbers grow and how processes scale efficiently. </li>

131 <li>Finance and Business:Log tables are also useful in finance and business for understanding growth, investment calculations, and<a>comparing</a>large numbers. They allow analysts and investors to perform calculations faster and understand trends more clearly.</li>

130 <li>Finance and Business:Log tables are also useful in finance and business for understanding growth, investment calculations, and<a>comparing</a>large numbers. They allow analysts and investors to perform calculations faster and understand trends more clearly.</li>

132 </ul><h3>Problem 1</h3>

131 </ul><h3>Problem 1</h3>

133 Find the value of log₁₀(37.28)

132 Find the value of log₁₀(37.28)

134 Okay, lets begin

133 Okay, lets begin

135 1.5730.

134 1.5730.

136 <h3>Explanation</h3>

135 <h3>Explanation</h3>

137 Step 1: Take the first four digits, 3728.

136 Step 1: Take the first four digits, 3728.

138 Step 2: In the log table, find the row for 37 and the column for 2.

137 Step 2: In the log table, find the row for 37 and the column for 2.

139 You get 0.5717.

138 You get 0.5717.

140 Step 3: The fourth digit is 8. The mean difference for 8 = 13.

139 Step 3: The fourth digit is 8. The mean difference for 8 = 13.

141 Add it: 5717 + 13 = 5730.

140 Add it: 5717 + 13 = 5730.

142 Step 4: Characteristic = (number of digits before the decimal - 1) = 1.

141 Step 4: Characteristic = (number of digits before the decimal - 1) = 1.

143 \(log10(37.28) = 1 + 0.5730 = 1.5730\)

142 \(log10(37.28) = 1 + 0.5730 = 1.5730\)

144 Well explained 👍

143 Well explained 👍

145 <h3>Problem 2</h3>

144 <h3>Problem 2</h3>

146 Find log₁₀(0.00452)

145 Find log₁₀(0.00452)

147 Okay, lets begin

146 Okay, lets begin

148 \(\log_{10}(0.00452) = -2.3447\).

147 \(\log_{10}(0.00452) = -2.3447\).

149 <h3>Explanation</h3>

148 <h3>Explanation</h3>

150 Step 1: First non-zero digits → 452.

149 Step 1: First non-zero digits → 452.

151 Step 2: Row = 45, column = 2 → table value = 0.6553.

150 Step 2: Row = 45, column = 2 → table value = 0.6553.

152 Step 3: Count zeros after the decimal before the first non-zero digit = 2.

151 Step 3: Count zeros after the decimal before the first non-zero digit = 2.

153 Characteristic\(= -(2 + 1) = -3\)

152 Characteristic\(= -(2 + 1) = -3\)

154 \(\log_{10}(0.00452) = -3 + 0.6553 = -2.3447\)

153 \(\log_{10}(0.00452) = -3 + 0.6553 = -2.3447\)

155 Well explained 👍

154 Well explained 👍

156 <h3>Problem 3</h3>

155 <h3>Problem 3</h3>

157 Find (15.2 × 42) / (3.6 × 28) using the logarithm table.

156 Find (15.2 × 42) / (3.6 × 28) using the logarithm table.

158 Okay, lets begin

157 Okay, lets begin

159 6.333.

158 6.333.

160 <h3>Explanation</h3>

159 <h3>Explanation</h3>

161 The logarithm of the given expression can be found using the logarithm properties.

160 The logarithm of the given expression can be found using the logarithm properties.

162 log (m / n) = log m - log n

161 log (m / n) = log m - log n

163 \(\log\frac{15.2 \times 42}{3.6 \times 28} = \log 15.2 + \log 42 - \log 3.6 - \log 28\)

162 \(\log\frac{15.2 \times 42}{3.6 \times 28} = \log 15.2 + \log 42 - \log 3.6 - \log 28\)

164 Next, use the log table to determine the log value.

163 Next, use the log table to determine the log value.

165 x

164 x

166 Characteristic Mantissa log x (characteristic + mantissa) 15.2

165 Characteristic Mantissa log x (characteristic + mantissa) 15.2

167 1

166 1

168 0.1818

167 0.1818

169 1.1818

168 1.1818

170 42

169 42

171 1

170 1

172 0.6232

171 0.6232

173 1.6232

172 1.6232

174 3.6

173 3.6

175 0

174 0

176 0.5563

175 0.5563

177 0.5563

176 0.5563

178 28

177 28

179 1

178 1

180 0.4472

179 0.4472

181 1.4472

180 1.4472

182 Now, we can substitute the values:

181 Now, we can substitute the values:

183 \(= (1.1818 + 1.6232) - (0.5563 + 1.4472) \)

182 \(= (1.1818 + 1.6232) - (0.5563 + 1.4472) \)

184 \(= 2.8050 - 2.0035 = 0.8015.\)

183 \(= 2.8050 - 2.0035 = 0.8015.\)

185 Next, use the anti-log table to determine the antilog of 0.8015.

184 Next, use the anti-log table to determine the antilog of 0.8015.

186 Antilog (0.8015) = 6.333

185 Antilog (0.8015) = 6.333

187 Thus, \(\frac{15.2 \times 42}{3.6 \times 28} = 6.333\)

186 Thus, \(\frac{15.2 \times 42}{3.6 \times 28} = 6.333\)

188 Well explained 👍

187 Well explained 👍

189 <h3>Problem 4</h3>

188 <h3>Problem 4</h3>

190 Multiply 23.7 × 0.452 Using Logs

189 Multiply 23.7 × 0.452 Using Logs

191 Okay, lets begin

190 Okay, lets begin

192 3. 7372.

191 3. 7372.

193 <h3>Explanation</h3>

192 <h3>Explanation</h3>

194 Find log₁₀(23.7) → Characteristic 1 + Mantissa 0.3747 → 1.3747

193 Find log₁₀(23.7) → Characteristic 1 + Mantissa 0.3747 → 1.3747

195 Find log₁₀(0.452) → Characteristic -1 + Mantissa 0.6553 → -0.3447

194 Find log₁₀(0.452) → Characteristic -1 + Mantissa 0.6553 → -0.3447

196 Add logs → \(1.3747 + (-0.3447) = 1.0300\)

195 Add logs → \(1.3747 + (-0.3447) = 1.0300\)

197 Find antilog → \(10^{1.0300} = 10 \times 10^{0.0300} \approx 10 \times 1.074 = 10.74\).

196 Find antilog → \(10^{1.0300} = 10 \times 10^{0.0300} \approx 10 \times 1.074 = 10.74\).

198 Well explained 👍

197 Well explained 👍

199 <h3>Problem 5</h3>

198 <h3>Problem 5</h3>

200 Find (12.6)² using logs

199 Find (12.6)² using logs

201 Okay, lets begin

200 Okay, lets begin

202 2.162.

201 2.162.

203 <h3>Explanation</h3>

202 <h3>Explanation</h3>

204 We use:

203 We use:

205 log (mn) = n log m

204 log (mn) = n log m

206 \(\log_{10}(12.6) = 1.1004\)

205 \(\log_{10}(12.6) = 1.1004\)

207 So, \(\log_{10}\big((12.6)^2\big) = 2 \times 1.1004 = 2.2008\)

206 So, \(\log_{10}\big((12.6)^2\big) = 2 \times 1.1004 = 2.2008\)

208 Antilog of \(2.2008 = 10^{0.2008} \times 10^2 = 1.59 \times 100\)

207 Antilog of \(2.2008 = 10^{0.2008} \times 10^2 = 1.59 \times 100\)

209 \((12.6)^2 \approx 159\).

208 \((12.6)^2 \approx 159\).

210 Well explained 👍

209 Well explained 👍

211 <h2>FAQs on Log Table</h2>

210 <h2>FAQs on Log Table</h2>

212 <h3>1.What do you mean by a log table?</h3>

211 <h3>1.What do you mean by a log table?</h3>

213 Before the advent of calculators and digital tools, mathematicians used log tables to solve complex mathematical problems. It is a mathematical chart to find the logarithms of numbers with the log base 10.

212 Before the advent of calculators and digital tools, mathematicians used log tables to solve complex mathematical problems. It is a mathematical chart to find the logarithms of numbers with the log base 10.

214 <h3>2.How to calculate the logarithm of a given number using a log table?</h3>

213 <h3>2.How to calculate the logarithm of a given number using a log table?</h3>

215 Step 1: Find the characteristic Step 2: Find the mantissa Step 3: Add the characteristic and mantissa together to get the logarithm.

214 Step 1: Find the characteristic Step 2: Find the mantissa Step 3: Add the characteristic and mantissa together to get the logarithm.

216 <h3>3.How to find the characteristic of a number?</h3>

215 <h3>3.How to find the characteristic of a number?</h3>

217 If the number is greater than 1, the formula for finding the characteristic is: Characteristic = the number of digits on the left side of the decimal point - 1 If the number is smaller than 1, the formula is: Characteristic = - (the number of zeros immediately followed by the decimal point + 1

216 If the number is greater than 1, the formula for finding the characteristic is: Characteristic = the number of digits on the left side of the decimal point - 1 If the number is smaller than 1, the formula is: Characteristic = - (the number of zeros immediately followed by the decimal point + 1

218 <h3>4.How to calculate the mantissa using a log table?</h3>

217 <h3>4.How to calculate the mantissa using a log table?</h3>

219 Step 1: Ignore the decimal point. Step 2: Use the first 4 significant digits of the number. Step 3: The first two digits give the row number, and the third digit gives the column number. Write the table value where the row and the column intersect in the log table. Step 4: The fourth digit is the mean difference, then add the mean difference to the value we get from step 3.

218 Step 1: Ignore the decimal point. Step 2: Use the first 4 significant digits of the number. Step 3: The first two digits give the row number, and the third digit gives the column number. Write the table value where the row and the column intersect in the log table. Step 4: The fourth digit is the mean difference, then add the mean difference to the value we get from step 3.

220 <h3>5.Can the mantissa be a negative value?</h3>

219 <h3>5.Can the mantissa be a negative value?</h3>

221 No, the mantissa is a decimal (fractional) part of a logarithm and is always positive. It can be found using the log table, which only gives positive decimal values. It remains positive even if the characteristic is negative.

220 No, the mantissa is a decimal (fractional) part of a logarithm and is always positive. It can be found using the log table, which only gives positive decimal values. It remains positive even if the characteristic is negative.

222 <h3>6.How can I check if my child is using the log table correctly?</h3>

221 <h3>6.How can I check if my child is using the log table correctly?</h3>

223 Ask them to solve simple multiplication or division problems using the log table and cross-check the answer manually or with a calculator. Ensure each step, reading the number, finding the log, adding/subtracting, then taking antilog, is correct.

222 Ask them to solve simple multiplication or division problems using the log table and cross-check the answer manually or with a calculator. Ensure each step, reading the number, finding the log, adding/subtracting, then taking antilog, is correct.

224 <h3>7.Should parents intervene if child struggles?</h3>

223 <h3>7.Should parents intervene if child struggles?</h3>

225 Yes, but gently. Ask guiding<a>questions</a>, break problems into manageable steps, and encourage them to reason instead of giving direct answers.

224 Yes, but gently. Ask guiding<a>questions</a>, break problems into manageable steps, and encourage them to reason instead of giving direct answers.

226 <h2>Jaskaran Singh Saluja</h2>

225 <h2>Jaskaran Singh Saluja</h2>

227 <h3>About the Author</h3>

226 <h3>About the Author</h3>

228 Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

227 Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

229 <h3>Fun Fact</h3>

228 <h3>Fun Fact</h3>

230 : He loves to play the quiz with kids through algebra to make kids love it.

229 : He loves to play the quiz with kids through algebra to make kids love it.