Fundamentals of Business Mathematics and Statistics · Indices and Logarithms

Introduction to Logarithms and Their Laws for CMA Foundation

Updated 10 October 2026 · Fact-checked

A logarithm is the power to which a base must be raised to get a number. If aˣ = N, then logₐN = x. To solve questions, convert between index and log form, then apply the product, quotient, power and change of base laws to simplify.

Understand Introduction to Logarithms and Their 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, so we write log₂8 = 3. It is the inverse of exponentiation, just as subtraction is the inverse of addition.

The index form and log form say the same thing. aˣ = N is the same as logₐN = x. Here a is the base, x is the logarithm and N is the number. The base must be positive and not equal to 1, and N must be positive. You cannot take the log of zero or of a negative number in this chapter.

Two results follow at once. logₐ1 = 0 because a⁰ = 1. logₐa = 1 because a¹ = a. Also a raised to logₐN equals N, since the log is the power that gives N.

Logarithms turn multiplication into addition, division into subtraction and powers into multiplication. That is why the laws below matter. They shrink hard calculations into easy ones.

Two bases are common. The common logarithm uses base 10 and is written log or log₁₀. The natural logarithm uses base e (about 2.718) and is written ln or logₑ. If a question gives no base, check the context. In this chapter it usually means base 10, and the laws work for any valid base.

Key formulas to remember

Index and log form: aˣ = N ⇔ logₐN = x; a > 0, a ≠ 1, N > 0.
Log of 1 and of the base: logₐ1 = 0 and logₐa = 1; True for every valid base a.
Product law: logₐ(mn) = logₐm + logₐn; m and n must both be positive.
Quotient law: logₐ(m ÷ n) = logₐm − logₐn; Order matters: numerator log first.
Power law: logₐ(mᵖ) = p × logₐm; Works for fractional p too, so log of a root is (1/n) × log m.
Change of base: logₐm = logᵦm ÷ logᵦa; Choose any new base b, usually 10.
Reciprocal (inverse) rule: logₐb = 1 ÷ log_b a; Put m = b and use b as the new base, so log_b b = 1.
Base-power rule: a^(logₐN) = N; The same base must appear in both places.
Common and natural logs: log = log₁₀ and ln = logₑ; Base 10 and base e, respectively.

How to solve Introduction to Logarithms and Their Laws questions

Use this method for any question on logarithms and their laws.

1Note the base and check that every number inside a log is positive.
2If the question mixes index and log forms, convert to one form using aˣ = N ⇔ logₐN = x.
3If the bases differ, bring them to one base with the change of base rule.
4Break the expression with the product, quotient and power laws, or combine separate logs into one.
5Use logₐ1 = 0 and logₐa = 1 to remove simple terms.
6Simplify the numbers and, if needed, convert back to index form to find the unknown.
7Substitute your answer back to check that each log still has a positive number.

Quickest way: Convert to index form and match powers

When to use it: Use when the options are neat numbers and the question asks for a value or an unknown in a single log equation.

Write the log as an index equation, for example logₐN = x becomes aˣ = N.
Express both sides as powers of the same prime, such as 2, 3 or 5.
Equate the powers and read off the answer.
For sums of logs, combine them into one log of a product, then find the value of the product.
If you are short on time, test each option in the index form.

Common mistakes in Introduction to Logarithms and Their Laws

Writing log(m + n) = log m + log n.
Students confuse the product law with addition.
Fix: The law is for a product only: log(mn) = log m + log n. There is no simple rule for log(m + n).
Treating log m ÷ log n as log(m ÷ n).
The quotient law and a ratio of logs look alike.
Fix: log(m ÷ n) = log m − log n. A ratio of two logs is the change of base form: log m ÷ log n = logₙm.
Applying the power law wrongly, as (log m)ᵖ = p log m.
The exponent position is misread.
Fix: The power goes on the number inside: log(mᵖ) = p log m. Squaring the whole log is a different thing.
Writing change of base upside down.
Students forget which log goes on top.
Fix: The number's log goes on top and the old base's log goes below: logₐm = log m ÷ log a.
Using logₐ1 = 1 or logₐa = 0.
The two results are swapped in memory.
Fix: Remember a⁰ = 1 gives logₐ1 = 0, and a¹ = a gives logₐa = 1.
Accepting answers that make a log of zero or a negative number.
Students solve the equation and skip the domain check.
Fix: Put the answer back into every log. Reject any value that makes the number inside a log zero or negative.

Worked examples

Example 1

Find the value of log₂32 + log₃9 − log₅5.

Show the solution

log₂32: since 2⁵ = 32, it equals 5.
log₃9: since 3² = 9, it equals 2.
log₅5 = 1, because the number equals the base.
Total = 5 + 2 − 1 = 6.

Answer: 6

Example 2

If log₁₀2 = 0.3010, find the value of log₁₀8 + log₁₀5 using the laws of logarithms.

Show the solution

log₁₀8 = log₁₀(2³) = 3 × log₁₀2 = 3 × 0.3010 = 0.9030.
log₁₀5 = log₁₀(10 ÷ 2) = log₁₀10 − log₁₀2 = 1 − 0.3010 = 0.6990.
Add: 0.9030 + 0.6990 = 1.6020.
Check by product law: log₁₀(8 × 5) = log₁₀40 = log₁₀4 + log₁₀10 = 0.6020 + 1 = 1.6020.

Answer: 1.6020

Exam tips

Memorise the laws (product, quotient, power, change of base) and the special values logₐ1 = 0 and logₐa = 1. Many MCQs need only one law applied correctly.
Look for hidden powers. Numbers like 8, 16, 27, 81 and 125 are the examiner's favourites.
When options are close, convert to index form and test them. This takes seconds and avoids sign slips.
Check the order in quotient and change of base questions, since wrong options are often the reversed form.
There is no negative marking, so attempt every question and eliminate options that break the basic rules.

Practice questions from Indices and Logarithms

Introduction to Logarithms and Their Laws in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

CA FoundationLogarithms: Definition and Laws

Introduction to Logarithms and Their Laws: frequently asked questions

What is the relation between a logarithm and an index?

They are two ways of writing one fact. If aˣ = N, then logₐN = x. The logarithm is simply the index (power) that the base needs to give the number.

What is the change of base formula for logarithms?

logₐm = logᵦm ÷ logᵦa, where b is any valid new base. The log of the number goes on top and the log of the old base goes below. It is used to compare or combine logs with different bases.

What is the difference between common log and natural log?

A common logarithm has base 10 and is written log or log₁₀. A natural logarithm has base e, about 2.718, and is written ln. All the laws of logarithms hold for both.

Can we take the logarithm of zero or a negative number?

No. The number inside a log must be positive, because a positive base raised to any real power never gives zero or a negative value. Always check this after solving an equation.