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CMA Foundation · Fundamentals of Business Mathematics and Statistics

Indices and Logarithms for CMA Foundation Paper 3

An index (power) shows how many times a base is multiplied by itself. A logarithm is the reverse: log_a N = x means aˣ = N. To solve questions, apply the index laws to bring everything to one base, or use the log laws to turn products and powers into sums and multiples.

What this chapter covers

This chapter has two linked parts. Indices cover powers such as 2⁵, negative and fractional powers, and the laws that let you simplify them. Logarithms are the same idea read backwards: if aˣ = N, then x = log_a N. The five topics build on each other, so the log laws are really the index laws in a new form.

In the objective paper you will mostly see short questions: simplify an expression, find x in an equation like 3ˣ = 81, evaluate a log such as log₂ 32, or state the characteristic of a common log. Each can be done in under a minute if the laws are automatic.

This chapter sits in Paper 3 (FBMS) with other algebra topics. Its skills, such as handling powers, changing form and solving for an unknown, are used again in equations, progressions and compound interest, where powers appear all the time. Getting the basics firm here saves time elsewhere in the paper.

Indices and logarithms are scoring because the questions are rule-based: if you know the laws, the answer follows in a few steps, and with no negative marking you can always attempt every question. The chapter is also short compared with its payoff. A few days of focused practice can make it one of your most reliable sections, and the same skills help in other Paper 3 chapters. Weak handling of powers, on the other hand, causes slips that spread into other questions.

Indices and Logarithms: topics in the order to study them

  1. 1Laws of IndicesEverything else in the chapter rests on these laws, so learn them first, including zero, negative and fractional powers.
  2. 2Simplifying Expressions and Solving Index EquationsApply the laws straight away: reduce to a common base and equate the powers. This builds the habit you need for logs.
  3. 3Introduction to Logarithms and Their LawsOnce indices are solid, a logarithm is just the power written another way, and each log law mirrors an index law.
  4. 4Solving Problems Using LogarithmsNow use the log laws and the definition to evaluate expressions, find unknowns and change base.
  5. 5Common Logarithms: Characteristic and MantissaThis is the most specific part, built on base 10 logs, so it comes last when the basics are secure.

How to prepare Indices and Logarithms

Treat this as a rules-and-practice chapter. Understand each law once, then drill until you apply it without thinking.

  1. Write the index laws on one page from memory, then check them. Repeat until you make no errors, including a⁰ = 1 and a⁻ⁿ = 1 ÷ aⁿ.
  2. Practise converting numbers to a common base, such as 8 = 2³, 27 = 3³, 125 = 5³, and 16 = 2⁴. Most index equations need this.
  3. For index equations, reduce both sides to the same base, then equate the powers and solve the simple equation.
  4. Learn the definition first: log_a N = x means aˣ = N. Convert between the two forms in both directions until it is quick.
  5. Write the log laws next to their index laws: log(mn) = log m + log n, log(m ÷ n) = log m − log n, log mⁿ = n log m. Note that log_a a = 1 and log_a 1 = 0.
  6. Practise the base-change rule, log_a b = log b ÷ log a, and the result log_a b × log_b a = 1, with small examples.
  7. For common logs, practise finding the characteristic from the position of the decimal point, and remember that the mantissa is always the positive decimal part.
  8. Finish with timed sets of 15 to 20 MCQs. Use elimination: test each option by substituting it back into the equation.

Common mistakes in Indices and Logarithms

  • Adding the powers when multiplying different bases, such as writing 2³ × 3² = 6⁵.

    Fix: Before adding powers, check the bases match. If they do not, evaluate the numbers or change to a common base.

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

    Fix: The log splits only products and quotients. log(m + n) has no simple expansion, so leave it or compute the sum first.

  • Treating a⁻ⁿ as a negative number.

    Fix: A negative power means a reciprocal. 2⁻³ = 1 ÷ 8, which is positive.

  • Equating powers when the bases are different, such as 4ˣ = 8 giving x = 2.

    Fix: Convert first: 4ˣ = 2²ˣ and 8 = 2³, so 2x = 3 and x = 3 ÷ 2.

  • Confusing log m ÷ log n with log(m ÷ n).

    Fix: log(m ÷ n) = log m − log n. log m ÷ log n (same base) equals log_n m, which is not log(m ÷ n).

  • Giving a negative mantissa when the log of a number is negative.

    Fix: Keep the mantissa positive. If log N = −1.3, write it as 2̄.7 (characteristic −2, mantissa 0.7), because −1.3 = −2 + 0.7.

Last-day revision: Indices and Logarithms

  • aᵐ × aⁿ = aᵐ⁺ⁿ and aᵐ ÷ aⁿ = aᵐ⁻ⁿ (same base).
  • (aᵐ)ⁿ = aᵐⁿ, and (ab)ⁿ = aⁿbⁿ.
  • a⁰ = 1 for a ≠ 0, and a⁻ⁿ = 1 ÷ aⁿ.
  • a^(1/n) is the nth root of a, and a^(m/n) = (ⁿ√a)ᵐ.
  • For an index equation, make the bases equal, then equate the powers.
  • log_a N = x means aˣ = N, defined for a > 0, a ≠ 1 and N > 0.
  • log_a a = 1 and log_a 1 = 0.
  • log(mn) = log m + log n, and log(m ÷ n) = log m − log n.
  • log mⁿ = n log m, and log_a b = log b ÷ log a.
  • log_a b × log_b a = 1.
  • Common log has base 10; the log of a number is the characteristic plus the mantissa.
  • The mantissa is always positive and lies from 0 up to but not including 1.

Indices and Logarithms practice questions

Indices and Logarithms in other exams

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

Indices and Logarithms: frequently asked questions

Is Indices and Logarithms difficult for CMA Foundation?

It is one of the more manageable chapters because it runs on a small set of rules. Most errors come from misapplying a law, not from hard concepts. Regular practice makes the questions quick.

Do I need to memorise log tables?

Focus on understanding the laws and the idea of characteristic and mantissa. Questions usually give any needed log values or ask about the structure of a log. Check the question for any values it supplies.

How should I attempt these MCQs in the exam?

Convert to a common base or apply the log law, then solve in one or two lines. If stuck, substitute each option back into the equation. Since there is no negative marking, always mark an answer.

Which topic should I study first?

Start with the laws of indices, because the log laws are built from them. Then do index equations before moving to logarithms.