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Fundamentals of Business Mathematics and Statistics · Indices and Logarithms

Solving Problems Using Logarithms: Equations and Identities

Updated 10 October 2026 · Fact-checked

Solving problems using logarithms means applying the log laws to simplify expressions, prove identities or find an unknown in a log equation. Combine or split logs using the product, quotient and power laws, convert to index form, solve, then check that every log argument is positive.

Understand Solving Problems Using Logarithms

A logarithm answers one question: to what power must the base be raised to get a number? If 2³ = 8, then log₂ 8 = 3. Every log problem is this idea written in a different way.

The log laws turn multiplication into addition, division into subtraction and powers into multiplication. Problems in this topic are mostly about choosing the right law in the right direction. To simplify, you often split or combine logs. To prove an identity, you work on one side until it matches the other.

A log equation has an unknown inside the log or in the base. The standard route is to collapse everything into a single log on each side. Then either equate the arguments or convert to index form.

One rule protects your marks: the argument of a log must be positive, and the base must be positive and not equal to 1. After solving, always test your answer against this. Roots that make an argument zero or negative must be rejected.

Key formulas to remember

Definition
logₐ N = x ⇔ aˣ = N
For a > 0, a ≠ 1, N > 0. Use it to switch between log form and index form.
Product law
logₐ (mn) = logₐ m + logₐ n
For m, n > 0.
Quotient law
logₐ (m ÷ n) = logₐ m − logₐ n
For m, n > 0.
Power law
logₐ (mᵖ) = p × logₐ m
For m > 0.
Change of base
logₐ m = logᵦ m ÷ logᵦ a
Also logₐ m = 1 ÷ logₘ a. Useful for chain and reciprocal problems.
Special values
logₐ 1 = 0; logₐ a = 1; a^(logₐ m) = m
The last result holds for m > 0.
Equal logs
If logₐ m = logₐ n, then m = n
Same base on both sides, with m, n > 0.

How to solve Solving Problems Using Logarithms questions

Use this method for equations, simplifications and identities.

  1. 1Write down the domain: every argument must be greater than 0, and any base must be positive and not 1.
  2. 2Convert any coefficient in front of a log into a power using the power law.
  3. 3Bring all logs to the same base. Use change of base if needed.
  4. 4Combine logs on each side into a single log using the product and quotient laws.
  5. 5Either equate the arguments (if both sides are logs of the same base) or convert to index form (if one side is a number).
  6. 6Solve the resulting algebraic or quadratic equation.
  7. 7Check each root against the domain and reject any that makes an argument zero or negative.
  8. 8For identities, start from the more complicated side and simplify until it equals the other side.

Quickest way: Convert to index form and test the options

When to use it: Use when the question is an MCQ with four numerical options for the unknown.

  1. If the equation is logₐ (expression) = number, write it as expression = aⁿᵘᵐᵇᵉʳ at once.
  2. Solve the simple equation that results.
  3. If the algebra looks long, substitute each option into the original equation and see which one works.
  4. Reject any option that makes an argument negative or zero.
  5. For value questions, replace each log by its known value (for example log₂ 8 = 3) rather than expanding further.

Common mistakes in Solving Problems Using Logarithms

  • Writing log (m + n) = log m + log n.

    The product law is confused with addition.

    Fix: The law splits a product, not a sum: log (mn) = log m + log n. There is no rule for log (m + n).

  • Treating (log m) ÷ (log n) as log (m ÷ n).

    The quotient law looks similar.

    Fix: log (m ÷ n) = log m − log n. A ratio of two logs is a change of base, logₙ m.

  • Not rejecting roots that make an argument negative.

    Students stop after solving the quadratic.

    Fix: Substitute every root back into each log. Keep only roots for which all arguments are positive.

  • Dropping the coefficient when combining logs, such as 2 log x becoming log 2x.

    The power law is forgotten.

    Fix: Move the coefficient up first: 2 log x = log x².

  • Mixing bases when adding or equating logs.

    Laws only work for one common base.

    Fix: Convert to a common base first, or use the change of base formula.

Worked examples

Example 1

Solve for x: log₂ (x + 2) + log₂ (x − 2) = 5.

Show the solution
  1. Domain: x + 2 > 0 and x − 2 > 0, so x > 2.
  2. Combine using the product law: log₂ [(x + 2)(x − 2)] = 5.
  3. Convert to index form: x² − 4 = 2⁵ = 32.
  4. So x² = 36 and x = 6 or x = −6.
  5. Check the domain: x = −6 is not greater than 2, so reject it.
  6. Verify x = 6: log₂ 8 + log₂ 4 = 3 + 2 = 5. Correct.

Answer: x = 6

Example 2

Prove that log (75 ÷ 16) − 2 log (5 ÷ 9) + log (32 ÷ 243) = log 2.

Show the solution
  1. Use the power law: 2 log (5 ÷ 9) = log (25 ÷ 81).
  2. The left side becomes log (75 ÷ 16) − log (25 ÷ 81) + log (32 ÷ 243).
  3. Combine into one log: log [(75 ÷ 16) × (81 ÷ 25) × (32 ÷ 243)].
  4. Multiply the first two: (75 × 81) ÷ (16 × 25) = 6075 ÷ 400 = 243 ÷ 16.
  5. Multiply by 32 ÷ 243: (243 × 32) ÷ (16 × 243) = 2.
  6. So the left side = log 2, which equals the right side.

Answer: Left side = log 2 = right side, so the identity is proved.

Exam tips

  • Convert to index form first in MCQs. It is usually the shortest route to the answer.
  • Always run the domain check, because one option is often the rejected root.
  • For value questions, break numbers into prime factors, such as 72 = 2³ × 3², and apply the laws.
  • In identities, use the power law first to remove coefficients, then combine into a single log.
  • There is no negative marking, so if time is short, test the options and choose the one that fits.

Practice questions from Indices and Logarithms

Solving Problems Using Logarithms: frequently asked questions

How do I solve a logarithmic equation step by step?

Note the domain, combine the logs into one on each side, then equate the arguments or convert to index form. Solve the equation that results. Finally, reject any root that makes a log argument zero or negative.

How do I prove a logarithmic identity?

Start with the more complicated side. Use the power law to clear coefficients, then the product and quotient laws to combine. Simplify the single argument until it matches the other side.

Why must I check my answers in log equations?

Logs are defined only for positive arguments. Squaring or expanding can produce roots that break this condition. Those roots are extraneous and must be rejected.

Can I use log (a + b) = log a + log b?

No. The product law applies only to log (ab). There is no simple law for the log of a sum.