Log Rules
2026-02-28 10:05 Diff
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Last updated on December 15, 2025

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.

What are Log Rules?

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

  • Product Rule: \(log_b(mn) = log_b m + log_b n \)
  • Quotient Rule: \(log_b{({m\over n})} = log_b m - log_b n\)
  • Power Rule:  \(log_b{({m^n})} = n log_b m\)
  • Change of Base: \(log_a b = {{log_c b \over log_c a}}\)

These guidelines are particularly helpful for solving logarithmic equations and simplifying complex logarithmic expressions. 

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

  • Since \(b^0=1\), we get \(log_b1=0\)
  • Since \(b^1=b\), the result will be \(log_bb=1\)

Working with logarithms in algebra and more complex mathematics is based on these fundamental principles.
 

What are Logarithm Rules

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

What are the Laws of Logarithms?


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.

Product Law

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

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

Quotient Law

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

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).

Power Law 

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

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

Change of Base Law

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

Because calculators usually only have buttons for logarithms in base 10 (common logarithms, represented by the symbol 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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How to Solve Logarithmic Equations?

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

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

1: Evaluate Logs
To evaluate logs, if the unknown variable is outside the logarithm, you should use the base of the logarithm to find the value of the logarithm itself.

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 argument are simple, but complex values may require a calculator.

Step 2: Convert to Exponential Form
When the unknown variable is located within the logarithm, rewrite the equation as an exponential expression.

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.

Step 3: Logarithm Combination
When an equation contains multiple logarithms, attempt to combine them by utilizing the properties of logarithms.

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.

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.

How to Change of Base Rule for Logarithms 

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 formula:

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

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.

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 solving equations, simplifying expressions, and evaluating non-standard bases.

Natural Log Rules

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.

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

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

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

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

\(In e=1\)

\(In 1 = 0\)

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

Product Rule of Logarithms

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

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

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

\(log_bm=x\) and \( log_bn=y\)

Converting these into exponential form, we get
 

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

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

Multiplying equations (1) and (2).

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


By using the laws of exponents,
 

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

Now converting back to logarithmic form:

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

Substituting values of x and y:

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

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

\(log(3a)=log 3+log a\)
 

\(log 10=log(5 × 2)=log 5+log 2\)
 

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

Quotient Rule of Logarithms

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

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

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

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

Converting these into exponential form, we get

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

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

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

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

By using the quotient rule of exponents

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

Now converting back to logarithmic form

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

Substituting the values of x and y:

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

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

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

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

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

Logarithm Power Rule

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

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

Let us derive,
 

Assume that,

\(\ \log_b m = x \ \)

Converting this to exponential form gives

\(\ b^x = m \ \)

Now raise both sides to the power n:

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

Using the power rule of exponents.

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

Converting back to logarithmic form:

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

Substituting

\(\ x = \log_b m \ \)
 

\(\ \log_b(m^n) = n \log_b m \ \)
 

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

\(\ \log(3^z) = z \log 3 \ \)
 

\(\ \log(y^2) = 2 \log y \ \)
 

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

Important Notes

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.
 

\(\ \log_a 1 = 0 \ \)

The most commonly used logarithmic rules are:

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

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

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

Tips and Tricks to Master Log Rules

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. 

  • A logarithm is used to find the power needed to get a number from its base. For example, \({log_{2} 8 = 3}\) because \({2^3} = 8\).  
  • When solving a complex logarithmic problem, rewrite the logarithmic expression in its equivalent exponential form. \(\log_b a = c \iff b^c = a\).
  • 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: \(\log_b (m^n) = n \log_b m\)
  • Use the change of base rule, if your calculator only supports base 10 or e. Use the formula \(\log_b a = \frac{\log a}{\log b}.\)
  • When solving logarithmic expressions, always solve them step by step. Apply one logarithmic rule at a time to ensure clarity, maintain accuracy, and minimize errors.
  • Parents can explain to children that a logarithm helps determine the power needed to raise a number to its base, using simple examples like powers of 10.
  • Teachers encourage students to rewrite logarithmic expressions in their equivalent exponential forms to understand better and solve complex problems.
  • Children always solve logarithmic expressions step by step and apply one rule at a time to avoid mistakes.

Common Mistakes and How to Avoid Them in Log Rules

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.

Real-Life Applications in Log Rules

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. 

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  • 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.
  • 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. 
  • In finance, logarithms are used to find how long it takes for an investment to grow. 
  • Logarithm is used to follow the exponential growth 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.
  • In chemistry, pH scale uses logarithms are used to measure acidity and alkalinity. 

Problem 1

log_10 (5 × 2)

Okay, lets begin

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

Explanation

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

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

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

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

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

Well explained 👍

Problem 2

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

Okay, lets begin

2
 

Explanation

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


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

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

Well explained 👍

Problem 3

Simply log_3 (9)^2

Okay, lets begin

4
 

Explanation

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

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

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

Well explained 👍

Problem 4

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

Okay, lets begin

 3

Explanation

Firstly, we will use the change of

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

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

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

Step 3: Finally, divide the values. 

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

The final answer is 3.
 

Well explained 👍

Problem 5

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

Okay, lets begin

12 digits
 

Explanation

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

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

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


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

Therefore, the answer is 12.
 

Well explained 👍

FAQs on Log Rules

1.What are logarithms?

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. 
 

2.How might I simplify phrases using these guidelines?

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

3.What is the natural logarithm?

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

4.Can a logarithm's base be negative?

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.
 

5.How can I work through logarithmic equations?

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

Jaskaran Singh Saluja

About the Author

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.

Fun Fact

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