Quantitative Aptitude · Ratio and Proportion, Indices and Logarithms

Logarithms: Definition and Laws for CA Foundation

Updated 1 October 2026 · Fact-checked

A logarithm is the power to which a base must be raised to get a number. If aˣ = N, then logₐ N = x, with a > 0, a ≠ 1 and N > 0. To solve questions, convert to the other form or apply the product, quotient and power laws.

Understand Logarithms: Definition and Laws

A logarithm answers one question: to what power must I raise the base to get this number? Since 2³ = 8, the power is 3. We write this as log₂ 8 = 3. Read it as "log of 8 to the base 2 is 3".

So exponential form and log form say the same thing. In aˣ = N, a is the base, x is the power and N is the number. In log form this becomes logₐ N = x. The base stays the base. The power becomes the answer of the log.

There are conditions. The base a must be positive and not equal to 1. The number N must be positive. You cannot take the log of zero or of a negative number, because no real power of a positive base gives such a value.

The laws of logarithms come straight from the laws of indices. Multiplying numbers means adding powers, so the log of a product is a sum of logs. Dividing means subtracting powers. Raising to a power means multiplying the power. Learn them as index laws in disguise and you will not forget them.

In Paper 3 these laws are used to simplify expressions, find unknown values and prove identities. Questions are short, so accuracy and speed matter.

Key formulas to remember

Definition: aˣ = N ⇔ logₐ N = x; Valid for a > 0, a ≠ 1, N > 0.
Log of 1: logₐ 1 = 0; Because a⁰ = 1.
Log of the base: logₐ a = 1; Because a¹ = a.
Product law: logₐ (mn) = logₐ m + logₐ n; m and n must both be positive.
Quotient law: logₐ (m ÷ n) = logₐ m − logₐ n; m and n must both be positive.
Power law: logₐ (mⁿ) = n × logₐ m; m must be positive.
Inverse relation: a^(logₐ N) = N; Useful for quick simplification.
Reciprocal: logₐ b = 1 ÷ log_b a; Also written logₐ b × log_b a = 1.

How to solve Logarithms: Definition and Laws questions

Use this method for any question on definition and laws of logarithms.

1Check the base and the number. The base must be positive and not 1. The number must be positive.
2If the question gives an exponential equation or asks for a log value, convert it to the other form using aˣ = N ⇔ logₐ N = x.
3Break the numbers into prime factors or powers of the base, such as 8 = 2³ or 81 = 3⁴.
4Apply the power law first to bring exponents down as multipliers.
5Use the product law to add logs and the quotient law to subtract logs. Combine into a single log if the option needs it.
6Substitute known values such as logₐ a = 1 and logₐ 1 = 0.
7Simplify the arithmetic and check the answer by converting back to exponential form.

Quickest way: Convert to powers of the base

When to use it: Use for value-finding and equation MCQs where the numbers are perfect powers.

Ask directly: base to what power gives the number? For log₃ 81, think 3 to what power is 81. Answer 4.
For an unknown base or number, rewrite in exponential form and compare.
For sums of logs with the same base, multiply the numbers inside. For differences, divide.
Test each option by converting it back to exponential form. This eliminates wrong options fast.
Skip questions with long unfamiliar bases unless a law reduces them in two steps. A wrong answer costs 0.25 marks.

Common mistakes in Logarithms: Definition and Laws

Writing log (m + n) = log m + log n.
Students confuse the product law with addition inside the log.
Fix: The sum of logs equals the log of the product: log m + log n = log (mn). There is no simple rule for log (m + n).
Writing log m ÷ log n as log (m ÷ n).
The quotient law looks similar.
Fix: The quotient law applies to log m − log n, not to a ratio of two logs. A ratio of logs is a change of base.
Writing log (mⁿ) = (log m)ⁿ.
The exponent is wrongly applied to the whole log.
Fix: The exponent comes down as a multiplier: log (mⁿ) = n log m.
Reading logₐ N = x and writing N = xᵃ.
Students mix up base and power while converting.
Fix: The base of the log is the base of the exponential: aˣ = N. Say it aloud: base to the power answer equals number.
Using a negative number, zero, or base 1 as valid.
The conditions are forgotten.
Fix: Always check a > 0, a ≠ 1 and N > 0 before accepting a solution, especially in equations in x.

Worked examples

Example 1

The value of log₂ 32 + log₃ 81 − log₅ 125 is: (a) 4 (b) 5 (c) 6 (d) 7

Show the solution

32 = 2⁵, so log₂ 32 = 5.
81 = 3⁴, so log₃ 81 = 4.
125 = 5³, so log₅ 125 = 3.
Total = 5 + 4 − 3 = 6.

Answer: (c) 6

Example 2

If log₁₀ 2 = a and log₁₀ 3 = b, then log₁₀ 72 in terms of a and b is: (a) 2a + 3b (b) 3a + 2b (c) 6a + b (d) a + b + 6

Show the solution

Factorise: 72 = 8 × 9 = 2³ × 3².
Apply the product law: log 72 = log 2³ + log 3².
Apply the power law: = 3 log 2 + 2 log 3.
Substitute: = 3a + 2b.

Answer: (b) 3a + 2b

Example 3

If log₄ x = 2.5, then x equals: (a) 10 (b) 16 (c) 32 (d) 64

Show the solution

Convert to exponential form: x = 4^2.5.
Write 2.5 as 5/2, so x = 4^(5/2).
4^(1/2) = 2, so x = 2⁵.
2⁵ = 32.

Answer: (c) 32

Exam tips

Most questions are value-finding. Write the number as a power of the base and read off the exponent.
When options look similar, convert each back to exponential form to check the one that fits.
Watch for traps involving log (m + n) or (log m)ⁿ, which are not valid laws.
In equations with an unknown inside a log, check that the final value keeps every log argument positive.
Do not spend long on one lengthy simplification. With 0.25 negative marking, skip and return if time remains.

Practice questions from Ratio and Proportion, Indices and Logarithms

Logarithms: Definition and Laws: frequently asked questions

What is the definition of a logarithm?

The logarithm of N to the base a is the power to which a must be raised to give N. It is written logₐ N = x, meaning aˣ = N. The base must be positive and not 1, and N must be positive.

How do I convert exponential form to logarithmic form?

Keep the base as the base of the log. Write the power as the answer and the result as the number. So 5³ = 125 becomes log₅ 125 = 3.

What are the three main laws of logarithms?

The product law says log (mn) = log m + log n. The quotient law says log (m ÷ n) = log m − log n. The power law says log (mⁿ) = n log m. All logs must have the same base and positive arguments.

Why is log of a negative number not defined?

A positive base raised to any real power always gives a positive result. So no real power produces a negative number or zero. Hence the log of such a number does not exist in real numbers.