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3 The rules that help to simplify and solve expressions involving logarithms are known as log rules. Log rules, being the inverse of exponent rules, are based on exponent properties. The rules of logarithms are used to expand or combine logarithmic expressions.

4 <h2>What are Log Rules?</h2>

5 What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math

6 ▶

7 Log rules are essential tools for manipulating and simplifying logarithmic<a>expressions</a>. The rules of<a>logarithms</a>are directly derived from the<a>rules of exponents</a>. Four primary logarithmic rules are commonly applied:

8 <ul><li>Product Rule: \(log_b(mn) = log_b m + log_b n \)</li>

9 </ul><ul><li>Quotient Rule: \(log_b{({m\over n})} = log_b m - log_b n\)</li>

10 </ul><ul><li>Power Rule: \(log_b{({m^n})} = n log_b m\)</li>

11 </ul><ul><li>Change of Base: \(log_a b = {{log_c b \over log_c a}}\)</li>

12 </ul>These guidelines are particularly helpful for solving logarithmic equations and simplifying complex logarithmic expressions.

13 Furthermore, the relationship between exponential and logarithmic forms \((b^x = m ⇔ log_bm=x)\) yields some basic identities:

14 <ul><li>Since \(b^0=1\), we get \(log_b1=0\)</li>

15 </ul><ul><li>Since \(b^1=b\), the result will be \(log_bb=1\)</li>

16 </ul>Working with logarithms in<a>algebra</a>and more complex mathematics is based on these fundamental principles.

17 <h2>What are Logarithm Rules</h2>

18 In<a>addition</a>to what we have seen already, there are a<a>number</a>of other logarithmic rules. The following table lists every logarithm rule:

19 <h2>What are the Laws of Logarithms?</h2>

20 Mathematical properties known as the laws of logarithms, or logarithmic rules, make logarithmic expressions and equations easier to understand and solve. Since logarithms are the opposite of exponentiation, these laws are predicated on the exponentiation rules.

21 Product Law

22 \({{{{log_{b} {(MN)} = {log_{b} M }+ {log_{b} N}}}}}\)

23 According to the first law, taking the logarithm after multiplying two numbers together is equivalent to adding their logarithms (of the same<a>base</a>).

24 Quotient Law

25 \({{{{{{log_{b} ({M \over N})}} = {{log_{b} M}} - {log_{b} N}}}}}\)

26 According to the second law, dividing two numbers and taking the logarithm of the result is the same as subtracting their logarithms (again, of the same base).

27 Power Law

28 \({{{{{{log_{b} ({M^{p})} = {{p \cdot log_{b}}} M}}}}}}\)

29 When a number is raised to a power, the logarithm is equal to the<a>product</a>of the<a>exponent</a>with the logarithm of the base expression.

30 Change of Base Law

31 \({{{{log_{a} b = {{{log_{c} b } \over {{log_{c}} a}}}}}}}\)

32 Because<a>calculators</a>usually only have buttons for logarithms in base 10 (common logarithms, represented by the<a>symbol</a>log) and base e (natural logarithms, represented by the symbol ln⁡), the change of base log law is especially helpful. However, you may come across logarithms in other bases in a variety of mathematical problems.

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35 <h2>How to Solve Logarithmic Equations?</h2>

34 <h2>How to Solve Logarithmic Equations?</h2>

36 Although log rules seem simple individually, they are often challenging in exams. Usually, you will have to use any number of<a>combinations</a>of<a>logarithmic functions</a>to derive an answer.

35 Although log rules seem simple individually, they are often challenging in exams. Usually, you will have to use any number of<a>combinations</a>of<a>logarithmic functions</a>to derive an answer.

37 Although at first this seems overwhelming, as always, the best technique to handle these issues is to break the<a>question</a>into smaller bits. Solving logarithmic problems uses algebraic manipulation and logarithmic properties. Here is a detailed guide broken out step-by-step:

36 Although at first this seems overwhelming, as always, the best technique to handle these issues is to break the<a>question</a>into smaller bits. Solving logarithmic problems uses algebraic manipulation and logarithmic properties. Here is a detailed guide broken out step-by-step:

38 1: Evaluate LogsTo evaluate logs, if the unknown<a>variable</a>is outside the logarithm, you should use the base of the logarithm to find the value of the logarithm itself.

37 1: Evaluate LogsTo evaluate logs, if the unknown<a>variable</a>is outside the logarithm, you should use the base of the logarithm to find the value of the logarithm itself.

39 For instance, if you have \({{( log_b(x) = y )}}\), you can express it as \({{{{{( x = b^y )}}}}}\) to find \({{{{( x )}}}}\). This is straightforward when the base and<a>argument</a>are simple, but complex values may require a calculator.

38 For instance, if you have \({{( log_b(x) = y )}}\), you can express it as \({{{{{( x = b^y )}}}}}\) to find \({{{{( x )}}}}\). This is straightforward when the base and<a>argument</a>are simple, but complex values may require a calculator.

40 Step 2: Convert to Exponential FormWhen the unknown variable is located within the logarithm, rewrite the<a>equation</a>as an exponential expression.

39 Step 2: Convert to Exponential FormWhen the unknown variable is located within the logarithm, rewrite the<a>equation</a>as an exponential expression.

41 For instance, if you have the equation \({{{{(\log_b(x) = y )}}}}\), you can rewrite it as \({{{{{(x = b^y )}}}}}\). You can find the unknown variable by taking the base of the logarithm and raising it to the power of the other side of the equation.

40 For instance, if you have the equation \({{{{(\log_b(x) = y )}}}}\), you can rewrite it as \({{{{{(x = b^y )}}}}}\). You can find the unknown variable by taking the base of the logarithm and raising it to the power of the other side of the equation.

42 Step 3: Logarithm CombinationWhen an equation contains<a>multiple</a>logarithms, attempt to combine them by utilizing the properties of logarithms.

41 Step 3: Logarithm CombinationWhen an equation contains<a>multiple</a>logarithms, attempt to combine them by utilizing the properties of logarithms.

43 For instance, two logarithms with the same base can be combined by addition or subtraction. As a result, the problem may become simpler. For example, the expression \({{{{{log_b}(x) + log_{b}(y)}}}}\) can be written as \({{{{{{log_{b}(x) + log_{b}(y) = log_{b}(xy)}}}}}}\) using the product rule.

42 For instance, two logarithms with the same base can be combined by addition or subtraction. As a result, the problem may become simpler. For example, the expression \({{{{{log_b}(x) + log_{b}(y)}}}}\) can be written as \({{{{{{log_{b}(x) + log_{b}(y) = log_{b}(xy)}}}}}}\) using the product rule.

44 Step 4: Look for extraneous solutions:Always verify the validity of the solution you have found. Because they lead to the undefined logarithm of a negative number or zero, some solutions might not be legitimate. These solutions are likely to be discarded as they are referred to as extraneous solutions.

43 Step 4: Look for extraneous solutions:Always verify the validity of the solution you have found. Because they lead to the undefined logarithm of a negative number or zero, some solutions might not be legitimate. These solutions are likely to be discarded as they are referred to as extraneous solutions.

45 <h2>How to Change of Base Rule for Logarithms </h2>

44 <h2>How to Change of Base Rule for Logarithms </h2>

46 When utilizing calculators that normally only support base 10 (common log) or base 𝑒 (natural log), the Change of Base Rule in logarithms enables you to transform a logarithm with one base into an equivalent expression with a different base. The following is the<a>formula</a>:

45 When utilizing calculators that normally only support base 10 (common log) or base 𝑒 (natural log), the Change of Base Rule in logarithms enables you to transform a logarithm with one base into an equivalent expression with a different base. The following is the<a>formula</a>:

47 \({{{log_{a} b} = {{log_c b} \over {log_c a}}}}\)

46 \({{{log_{a} b} = {{log_c b} \over {log_c a}}}}\)

48 In this case, the original base is denoted by 𝑎, the argument by 𝑏, and the new base by 𝑐, which can be any positive number other than 1.

47 In this case, the original base is denoted by 𝑎, the argument by 𝑏, and the new base by 𝑐, which can be any positive number other than 1.

49 For example, you may rewrite \({{log_2 8}}\) using base 10 as follows to evaluate it: \({{{log 8} \over {log 2}} \approx {{{0.9031} \over {0.3010}}} = {{3}}}\). Similarly, \({{log_3 9}}\), utilizing natural logs, will be \({{{{ln \text { 9}} \over {ln \text { 3}}} \approx {{2.1972} \over {1.0986}} = 2}}\). This rule is useful for<a>solving equations</a>,<a>simplifying expressions</a>, and evaluating non-standard bases.

48 For example, you may rewrite \({{log_2 8}}\) using base 10 as follows to evaluate it: \({{{log 8} \over {log 2}} \approx {{{0.9031} \over {0.3010}}} = {{3}}}\). Similarly, \({{log_3 9}}\), utilizing natural logs, will be \({{{{ln \text { 9}} \over {ln \text { 3}}} \approx {{2.1972} \over {1.0986}} = 2}}\). This rule is useful for<a>solving equations</a>,<a>simplifying expressions</a>, and evaluating non-standard bases.

50 <h2>Natural Log Rules</h2>

49 <h2>Natural Log Rules</h2>

51 A natural logarithm is a logarithm with base e and is written as In. That is, loge=In. We do not usually write the base for natural logarithm because whenever we see In, it is automatically understood that the base is e. The laws of logarithms apply to natural logarithms in the same way they do to all other logarithms. The important rules of natural logarithms are listed below.

50 A natural logarithm is a logarithm with base e and is written as In. That is, loge=In. We do not usually write the base for natural logarithm because whenever we see In, it is automatically understood that the base is e. The laws of logarithms apply to natural logarithms in the same way they do to all other logarithms. The important rules of natural logarithms are listed below.

52 \(In(mn)=In m+In n\)

51 \(In(mn)=In m+In n\)

53 \(In(\frac{m}{n})=In m-In n\)

52 \(In(\frac{m}{n})=In m-In n\)

54 \(In(m^n)=n In m\)

53 \(In(m^n)=n In m\)

55 \(In a=\frac{log a}{log e}\)

54 \(In a=\frac{log a}{log e}\)

56 \(In e=1\)

55 \(In e=1\)

57 \(In 1 = 0\)

56 \(In 1 = 0\)

58 The “number raised to log” rule is \(b^{log_b^x}=x\) The equivalent rule for natural logarithms is \(e^{In x}=x\).

57 The “number raised to log” rule is \(b^{log_b^x}=x\) The equivalent rule for natural logarithms is \(e^{In x}=x\).

59 <h2>Product Rule of Logarithms</h2>

58 <h2>Product Rule of Logarithms</h2>

60 According to the product rule of logarithms, the logarithm of the product of two numbers is equal to the<a>sum</a>of the logarithms of each number. In other words,

59 According to the product rule of logarithms, the logarithm of the product of two numbers is equal to the<a>sum</a>of the logarithms of each number. In other words,

61 \(log_b(mn)=log_bm+log_bn\)

60 \(log_b(mn)=log_bm+log_bn\)

62 Let us see how this rule is derived. Let us assume,

61 Let us see how this rule is derived. Let us assume,

63 \(log_bm=x\) and \( log_bn=y\)

62 \(log_bm=x\) and \( log_bn=y\)

64 Converting these into<a>exponential form</a>, we get

63 Converting these into<a>exponential form</a>, we get

65 \(m=b^x (1) \)

64 \(m=b^x (1) \)

66 \(n=b^y (2)\)

65 \(n=b^y (2)\)

67 Multiplying equations (1) and (2).

66 Multiplying equations (1) and (2).

68 \(mn=b^x. b^y\)

67 \(mn=b^x. b^y\)

69 By using the laws of exponents,

68 By using the laws of exponents,

70 \(mn=b^{x+y}\)

69 \(mn=b^{x+y}\)

71 Now converting back to logarithmic form:

70 Now converting back to logarithmic form:

72 \(log_b(mn)=x+y\)

71 \(log_b(mn)=x+y\)

73 Substituting values of x and y:

72 Substituting values of x and y:

74 \(log_b(mn)=log_bm+log_bn\)

73 \(log_b(mn)=log_bm+log_bn\)

75 Thus, the product rule of logarithms is proved. This rule can be applied in examples such as.

74 Thus, the product rule of logarithms is proved. This rule can be applied in examples such as.

76 \(log(3a)=log 3+log a\)

75 \(log(3a)=log 3+log a\)

77 \(log 10=log(5 × 2)=log 5+log 2\)

76 \(log 10=log(5 × 2)=log 5+log 2\)

78 \(log_3(ab)=log_3a+log_3b\)

77 \(log_3(ab)=log_3a+log_3b\)

79 <h2>Quotient Rule of Logarithms</h2>

78 <h2>Quotient Rule of Logarithms</h2>

80 According to the<a>quotient</a>rule of logarithms, the logarithm of a quotient of two numbers is equal to the difference of the logarithms of those numbers. That is,

79 According to the<a>quotient</a>rule of logarithms, the logarithm of a quotient of two numbers is equal to the difference of the logarithms of those numbers. That is,

81 \(\ \log_b\left(\frac{m}{n}\right) = \log_b m - \log_b n \ \)

80 \(\ \log_b\left(\frac{m}{n}\right) = \log_b m - \log_b n \ \)

82 Let us derive this rule step by step. Let us assume that,

81 Let us derive this rule step by step. Let us assume that,

83 \(\ \log_b m = x \ \) and \(\ \log_b n = y \ \)

82 \(\ \log_b m = x \ \) and \(\ \log_b n = y \ \)

84 Converting these into exponential form, we get

83 Converting these into exponential form, we get

85 \(\ m = b^x \quad (1) \ \)

84 \(\ m = b^x \quad (1) \ \)

86 \(\ n = b^y \quad (2) \ \)

85 \(\ n = b^y \quad (2) \ \)

87 Dividing equation (1) and by equation (2).

86 Dividing equation (1) and by equation (2).

88 \(\ \frac{m}{n} = \frac{b^x}{b^y} \ \)

87 \(\ \frac{m}{n} = \frac{b^x}{b^y} \ \)

89 By using the<a>quotient rule of exponents</a>

88 By using the<a>quotient rule of exponents</a>

90 \(\ \frac{m}{n} = b^{x - y} \ \)

89 \(\ \frac{m}{n} = b^{x - y} \ \)

91 Now converting back to logarithmic form

90 Now converting back to logarithmic form

92 \(\ \log_b\left(\frac{m}{n}\right) = x - y \ \)

91 \(\ \log_b\left(\frac{m}{n}\right) = x - y \ \)

93 Substituting the values of x and y:

92 Substituting the values of x and y:

94 \(\ \log_b\left(\frac{m}{n}\right) = \log_b m - \log_b n \ \)

93 \(\ \log_b\left(\frac{m}{n}\right) = \log_b m - \log_b n \ \)

95 Thus, the quotient rule of logarithms is proved. This rule can be applied in examples such as

94 Thus, the quotient rule of logarithms is proved. This rule can be applied in examples such as

96 \(\ \log\left(\frac{y}{3}\right) = \log y - \log 3 \ \)

95 \(\ \log\left(\frac{y}{3}\right) = \log y - \log 3 \ \)

97 \(\ \log 25 = \log\left(\frac{125}{3}\right) = \log 125 - \log 3 \ \)

96 \(\ \log 25 = \log\left(\frac{125}{3}\right) = \log 125 - \log 3 \ \)

98 \(\ \log_7\left(\frac{a}{b}\right) = \log_7 a - \log_7 b \ \)

97 \(\ \log_7\left(\frac{a}{b}\right) = \log_7 a - \log_7 b \ \)

99 <h2>Logarithm Power Rule</h2>

98 <h2>Logarithm Power Rule</h2>

100 The power rule of logarithm states that when a logarithm contains a<a>term</a>raised to a power, the exponent can be moved in front of the logarithm.

99 The power rule of logarithm states that when a logarithm contains a<a>term</a>raised to a power, the exponent can be moved in front of the logarithm.

101 \(\ \log_b(m^n) = n \log_b m \ \)

100 \(\ \log_b(m^n) = n \log_b m \ \)

102 Let us derive,

101 Let us derive,

103 Assume that,

102 Assume that,

104 \(\ \log_b m = x \ \)

103 \(\ \log_b m = x \ \)

105 Converting this to exponential form gives

104 Converting this to exponential form gives

106 \(\ b^x = m \ \)

105 \(\ b^x = m \ \)

107 Now raise both sides to the power n:

106 Now raise both sides to the power n:

108 \(\ (b^x)^n = m^n \ \)

107 \(\ (b^x)^n = m^n \ \)

109 Using the power rule of exponents.

108 Using the power rule of exponents.

110 \(\ b^{nx} = m^n \ \)

109 \(\ b^{nx} = m^n \ \)

111 Converting back to logarithmic form:

110 Converting back to logarithmic form:

112 \(\ \log_b(m^n) = nx \ \)

111 \(\ \log_b(m^n) = nx \ \)

113 Substituting

112 Substituting

114 \(\ x = \log_b m \ \)

113 \(\ x = \log_b m \ \)

115 \(\ \log_b(m^n) = n \log_b m \ \)

114 \(\ \log_b(m^n) = n \log_b m \ \)

116 Thus, the power rule of logarithms is proved. Some examples of this rule are:

115 Thus, the power rule of logarithms is proved. Some examples of this rule are:

117 \(\ \log(3^z) = z \log 3 \ \)

116 \(\ \log(3^z) = z \log 3 \ \)

118 \(\ \log(y^2) = 2 \log y \ \)

117 \(\ \log(y^2) = 2 \log y \ \)

119 \(\ \log_3(y^x) = x \log_3 y \ \)

118 \(\ \log_3(y^x) = x \log_3 y \ \)

120 <h2>Important Notes</h2>

119 <h2>Important Notes</h2>

121 The laws of logarithms are the same for natural logarithms and common logarithms. The only difference is the base, which remains the same throughout when applying the rules. For any base 𝑎.

120 The laws of logarithms are the same for natural logarithms and common logarithms. The only difference is the base, which remains the same throughout when applying the rules. For any base 𝑎.

122 \(\ \log_a 1 = 0 \ \)

121 \(\ \log_a 1 = 0 \ \)

123 The most commonly used logarithmic rules are:

122 The most commonly used logarithmic rules are:

124 \(\ \log_b(mn) = \log_b m + \log_b n \ \)

123 \(\ \log_b(mn) = \log_b m + \log_b n \ \)

125 \(\ \log_b\left(\frac{m}{n}\right) = \log_b m - \log_b n \ \)

124 \(\ \log_b\left(\frac{m}{n}\right) = \log_b m - \log_b n \ \)

126 \(\ \log_b(m^n) = n \log_b m \ \)

125 \(\ \log_b(m^n) = n \log_b m \ \)

127 <h2>Tips and Tricks to Master Log Rules</h2>

126 <h2>Tips and Tricks to Master Log Rules</h2>

128 Mastering logarithms is important as it helps students to solve complex problems, simplify calculations, and understand real-life applications. Here we will learn some tips and tricks to master log rules.

127 Mastering logarithms is important as it helps students to solve complex problems, simplify calculations, and understand real-life applications. Here we will learn some tips and tricks to master log rules.

129 <ul><li>A logarithm is used to find the power needed to get a number from its base. For example, \({log_{2} 8 = 3}\) because \({2^3} = 8\). </li>

128 <ul><li>A logarithm is used to find the power needed to get a number from its base. For example, \({log_{2} 8 = 3}\) because \({2^3} = 8\). </li>

130 <li>When solving a complex logarithmic problem, rewrite the logarithmic expression in its equivalent exponential form. \(\log_b a = c \iff b^c = a\).</li>

129 <li>When solving a complex logarithmic problem, rewrite the logarithmic expression in its equivalent exponential form. \(\log_b a = c \iff b^c = a\).</li>

131 <li>Memorize the three main rules of logarithms. They are Product rule: \(\log_b (mn) = \log_b m + \log_b n\) Quotient rule: \(\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n\) Power rule: \(\log_b (m^n) = n \log_b m\)</li>

130 <li>Memorize the three main rules of logarithms. They are Product rule: \(\log_b (mn) = \log_b m + \log_b n\) Quotient rule: \(\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n\) Power rule: \(\log_b (m^n) = n \log_b m\)</li>

132 <li>Use the change of base rule, if your calculator only supports base 10 or e. Use the formula \(\log_b a = \frac{\log a}{\log b}.\)</li>

131 <li>Use the change of base rule, if your calculator only supports base 10 or e. Use the formula \(\log_b a = \frac{\log a}{\log b}.\)</li>

133 <li>When solving logarithmic expressions, always solve them step by step. Apply one logarithmic rule at a time to ensure clarity, maintain<a>accuracy</a>, and minimize errors.</li>

132 <li>When solving logarithmic expressions, always solve them step by step. Apply one logarithmic rule at a time to ensure clarity, maintain<a>accuracy</a>, and minimize errors.</li>

134 <li>Parents can explain to children that a logarithm helps determine the power needed to raise a number to its base, using simple examples like<a>powers of 10</a>.</li>

133 <li>Parents can explain to children that a logarithm helps determine the power needed to raise a number to its base, using simple examples like<a>powers of 10</a>.</li>

135 <li>Teachers encourage students to rewrite logarithmic expressions in their equivalent exponential forms to understand better and solve complex problems.</li>

134 <li>Teachers encourage students to rewrite logarithmic expressions in their equivalent exponential forms to understand better and solve complex problems.</li>

136 <li>Children always solve logarithmic expressions step by step and apply one rule at a time to avoid mistakes.</li>

135 <li>Children always solve logarithmic expressions step by step and apply one rule at a time to avoid mistakes.</li>

137 </ul><h2>Common Mistakes and How to Avoid Them in Log Rules</h2>

136 </ul><h2>Common Mistakes and How to Avoid Them in Log Rules</h2>

138 Common errors include misuse of rules, using zero or negative values, and incorrect handling of bases or exponents. By avoiding these mistakes, we can successfully solve logarithmic expressions and also build self-confidence over a period of time.

137 Common errors include misuse of rules, using zero or negative values, and incorrect handling of bases or exponents. By avoiding these mistakes, we can successfully solve logarithmic expressions and also build self-confidence over a period of time.

139 <h2>Real-Life Applications in Log Rules</h2>

138 <h2>Real-Life Applications in Log Rules</h2>

140 Logarithmic rules are not limited to mathematics, they are used in various real-life applications. In this section, we will learn some application of logarithmic rules.

139 Logarithmic rules are not limited to mathematics, they are used in various real-life applications. In this section, we will learn some application of logarithmic rules.

141 .

140 .

142 <ul><li>In geology, the Richter scale uses logarithms to measure the magnitude of earthquakes. As the scale is logarithmic, an increase of one unit represents a tenfold increase in wave amplitude and approximately 31.6 times more energy released.</li>

141 <ul><li>In geology, the Richter scale uses logarithms to measure the magnitude of earthquakes. As the scale is logarithmic, an increase of one unit represents a tenfold increase in wave amplitude and approximately 31.6 times more energy released.</li>

143 </ul><ul><li>Sound loudness is measures in decibels (dB) using logarithms. As human hearing responds to sound in a logarithmic way, the formula to represents how we perceive changes in loudness is: \({{L}} = {{log}_{10} {({I \over {I_{0}}})}}\). It is used in audio engineering, noise, and health safety. </li>

142 </ul><ul><li>Sound loudness is measures in decibels (dB) using logarithms. As human hearing responds to sound in a logarithmic way, the formula to represents how we perceive changes in loudness is: \({{L}} = {{log}_{10} {({I \over {I_{0}}})}}\). It is used in audio engineering, noise, and health safety. </li>

144 <li>In finance, logarithms are used to find how long it takes for an investment to grow. </li>

143 <li>In finance, logarithms are used to find how long it takes for an investment to grow. </li>

145 <li>Logarithm is used to follow the<a>exponential growth</a>patterns. So it is used in physics, biology, and environmental science to find how it will take for a quantity to reduce or grow to a certain level.</li>

144 <li>Logarithm is used to follow the<a>exponential growth</a>patterns. So it is used in physics, biology, and environmental science to find how it will take for a quantity to reduce or grow to a certain level.</li>

146 <li>In chemistry, pH scale uses logarithms are used to measure acidity and alkalinity. </li>

145 <li>In chemistry, pH scale uses logarithms are used to measure acidity and alkalinity. </li>

147 </ul><h3>Problem 1</h3>

146 </ul><h3>Problem 1</h3>

148 log_10 (5 × 2)

147 log_10 (5 × 2)

149 Okay, lets begin

148 Okay, lets begin

150 \({{log_{10} 10 = 1 }}\)

149 \({{log_{10} 10 = 1 }}\)

151 <h3>Explanation</h3>

150 <h3>Explanation</h3>

152 Break the multiplication into parts, use approximate values, and add them to get the final answer.

151 Break the multiplication into parts, use approximate values, and add them to get the final answer.

153 \({{Log_{10} (5 \times 2) = log_{10} 5 + log_{10} 2 }}\)

152 \({{Log_{10} (5 \times 2) = log_{10} 5 + log_{10} 2 }}\)

154 \({{log_{10} 5 \approx {0.6990}, {\text { and }} {log_{10} 2} \approx 0.3010 }}\)

153 \({{log_{10} 5 \approx {0.6990}, {\text { and }} {log_{10} 2} \approx 0.3010 }}\)

155 \({{0.6990 + 0.3010 = 1}}\)

154 \({{0.6990 + 0.3010 = 1}}\)

156 The final result is \({{log_{10} 10 = 1}}\)

155 The final result is \({{log_{10} 10 = 1}}\)

157 Well explained 👍

156 Well explained 👍

158 <h3>Problem 2</h3>

157 <h3>Problem 2</h3>

159 Solve using the Division Rule: log_2 (16/4)

158 Solve using the Division Rule: log_2 (16/4)

160 Okay, lets begin

159 Okay, lets begin

161 2

160 2

162 <h3>Explanation</h3>

161 <h3>Explanation</h3>

163 Apply the log rule. \({{Log_{2} {({16\over 4})} = {log_2 16} - {log_2 4}}}\)

162 Apply the log rule. \({{Log_{2} {({16\over 4})} = {log_2 16} - {log_2 4}}}\)

164 Step 2:Convert them to the power of 2. \({{16 = 2^4}}\), so \({{log_2 16 = 4}}\) Then, \({{2^2 = 4}}\), so \({{log_2 4 = 2}}\)

163 Step 2:Convert them to the power of 2. \({{16 = 2^4}}\), so \({{log_2 16 = 4}}\) Then, \({{2^2 = 4}}\), so \({{log_2 4 = 2}}\)

165 Step 3:Subtract the end results, that is, 2 from 4 \(4 - 2 = 2\)

164 Step 3:Subtract the end results, that is, 2 from 4 \(4 - 2 = 2\)

166 Well explained 👍

165 Well explained 👍

167 <h3>Problem 3</h3>

166 <h3>Problem 3</h3>

168 Simply log_3 (9)^2

167 Simply log_3 (9)^2

169 Okay, lets begin

168 Okay, lets begin

170 4

169 4

171 <h3>Explanation</h3>

170 <h3>Explanation</h3>

172 Use power rule \({{Log_3 (9)^2 = 2 log_3 9}}\)

171 Use power rule \({{Log_3 (9)^2 = 2 log_3 9}}\)

173 Step 2:We will simplify log3 9: \({{9 = 3^2}}\), so \({log_3 9 = log_3(3^2) = 2}\)

172 Step 2:We will simplify log3 9: \({{9 = 3^2}}\), so \({log_3 9 = log_3(3^2) = 2}\)

174 Step 3:Substitute \({log_3 9 = 2 }\)into the expression \({2 \cdot log_3 9 = 2 × 2 = 4 }\)

173 Step 3:Substitute \({log_3 9 = 2 }\)into the expression \({2 \cdot log_3 9 = 2 × 2 = 4 }\)

175 Well explained 👍

174 Well explained 👍

176 <h3>Problem 4</h3>

175 <h3>Problem 4</h3>

177 Convert log_4 64 to a common log (base 10).

176 Convert log_4 64 to a common log (base 10).

178 Okay, lets begin

177 Okay, lets begin

179 3

178 3

180 <h3>Explanation</h3>

179 <h3>Explanation</h3>

181 Firstly, we will use the change of

180 Firstly, we will use the change of

182 \({ Log_4 64 = {{log_{10} 64} \over {log_{10} 4}}}\)

181 \({ Log_4 64 = {{log_{10} 64} \over {log_{10} 4}}}\)

183 Step 2:Next we will find the logarithms using a base of 10 to be calculated:

182 Step 2:Next we will find the logarithms using a base of 10 to be calculated:

184 \({Log_{10} 64 \approx 1.8062}\) \({Log_{10} 4 \approx 0.6021} \)

183 \({Log_{10} 64 \approx 1.8062}\) \({Log_{10} 4 \approx 0.6021} \)

185 Step 3:Finally, divide the values.

184 Step 3:Finally, divide the values.

186 \({{1.8062 \over 0.6021} \approx 3}\)

185 \({{1.8062 \over 0.6021} \approx 3}\)

187 The final answer is 3.

186 The final answer is 3.

188 Well explained 👍

187 Well explained 👍

189 <h3>Problem 5</h3>

188 <h3>Problem 5</h3>

190 If log 3 = 0.477, find the number of digits in 3^(25).

189 If log 3 = 0.477, find the number of digits in 3^(25).

191 Okay, lets begin

190 Okay, lets begin

192 12 digits

191 12 digits

193 <h3>Explanation</h3>

192 <h3>Explanation</h3>

194 Utilize the formula to determine the number of N digits in any number:

193 Utilize the formula to determine the number of N digits in any number:

195 Number of digits in N = \({(log_{10} N) + 1}\)

194 Number of digits in N = \({(log_{10} N) + 1}\)

196 Let’s consider here \({N = 3^{25}} \) So, \({Log_{10} (3^{25}) = 25 \cdot log_{10} 3 }\) \({= 25 \cdot 0.477}\) \({= 11.925}\)

195 Let’s consider here \({N = 3^{25}} \) So, \({Log_{10} (3^{25}) = 25 \cdot log_{10} 3 }\) \({= 25 \cdot 0.477}\) \({= 11.925}\)

197 Now, apply the formula: Number of Digits = \({⌊11.925⌋ + 1 = 11 + 1 = 12}\)

196 Now, apply the formula: Number of Digits = \({⌊11.925⌋ + 1 = 11 + 1 = 12}\)

198 Therefore, the answer is 12.

197 Therefore, the answer is 12.

199 Well explained 👍

198 Well explained 👍

200 <h2>FAQs on Log Rules</h2>

199 <h2>FAQs on Log Rules</h2>

201 <h3>1.What are logarithms?</h3>

200 <h3>1.What are logarithms?</h3>

202 Logarithms are the inverse of exponents. They are used to find the precise power to which a base must be raised to produce the desired outcome.

201 Logarithms are the inverse of exponents. They are used to find the precise power to which a base must be raised to produce the desired outcome.

203 <h3>2.How might I simplify phrases using these guidelines?</h3>

202 <h3>2.How might I simplify phrases using these guidelines?</h3>

204 To simplify difficult equations for simpler assessment or solving, apply log rules, including product, quotient, and power rules.

203 To simplify difficult equations for simpler assessment or solving, apply log rules, including product, quotient, and power rules.

205 <h3>3.What is the natural logarithm?</h3>

204 <h3>3.What is the natural logarithm?</h3>

206 The natural logarithm is a type of logarithm that uses mathematics<a>constant</a>e (e = 2.718) as it base. It is written as ln(x) instead of \({log_e(x)}\).

205 The natural logarithm is a type of logarithm that uses mathematics<a>constant</a>e (e = 2.718) as it base. It is written as ln(x) instead of \({log_e(x)}\).

207 <h3>4.Can a logarithm's base be negative?</h3>

206 <h3>4.Can a logarithm's base be negative?</h3>

208 No, a logarithm's base has to be positive and not equal to 1. In real numbers, a negative base would render the log undefined.

207 No, a logarithm's base has to be positive and not equal to 1. In real numbers, a negative base would render the log undefined.

209 <h3>5.How can I work through logarithmic equations?</h3>

208 <h3>5.How can I work through logarithmic equations?</h3>

210 Separate the log term, then use log rules or change the equation to exponential form. Check answers to avoid extraneous solutions.

209 Separate the log term, then use log rules or change the equation to exponential form. Check answers to avoid extraneous solutions.

211 <h2>Jaskaran Singh Saluja</h2>

210 <h2>Jaskaran Singh Saluja</h2>

212 <h3>About the Author</h3>

211 <h3>About the Author</h3>

213 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.

212 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.

214 <h3>Fun Fact</h3>

213 <h3>Fun Fact</h3>

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

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