Indices and Logarithms: formula sheet

Key formulas

Product law

aᵐ × aⁿ = aᵐ⁺ⁿ

Same base only. Add the indices.

Quotient law

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Same base, a ≠ 0. Subtract the indices.

Power of a power

(aᵐ)ⁿ = aᵐⁿ

Multiply the indices.

Power of a product

(ab)ⁿ = aⁿ × bⁿ

The index goes to each factor.

Power of a quotient

(a/b)ⁿ = aⁿ / bⁿ

For b ≠ 0.

Zero index

a⁰ = 1

For a ≠ 0. 0⁰ is not defined at this level.

Negative index

a⁻ⁿ = 1/aⁿ

For a ≠ 0. Also (a/b)⁻ⁿ = (b/a)ⁿ.

Fractional index

a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ

Take the root first, then the power.

Product law

aᵐ × aⁿ = a^(m+n)

Same base only. Add the indices.

Quotient law

aᵐ ÷ aⁿ = a^(m−n)

Same base, a ≠ 0. Subtract the indices.

Power of a power

(aᵐ)ⁿ = a^(mn)

Multiply the indices.

Power of a product

(ab)ⁿ = aⁿ bⁿ

Also (a/b)ⁿ = aⁿ ÷ bⁿ for b ≠ 0.

Zero and negative index

a⁰ = 1 and a^(−n) = 1 ÷ aⁿ

For a ≠ 0.

Fractional index

a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ

Gives the link between surds and indices.

Equating powers

If aˣ = aʸ (a > 0, a ≠ 1), then x = y

The main tool for index equations.

Equating bases

If xⁿ = yⁿ with x, y > 0 and n ≠ 0, then x = y

Use when the indices are already equal. If n is odd, positivity is not needed. If n is even and signs are unrestricted, x = ±y.

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.

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.

Quick revision

• 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.

Common mistakes

• Adding indices when bases are different, such as 2³ × 3² = 6⁵. Fix: Check the bases first. If they differ, convert to a common base or evaluate separately: 8 × 9 = 72.
• Treating a⁻ⁿ as a negative number, so 2⁻³ = −8. Fix: A negative index means reciprocal: 2⁻³ = 1/2³ = 1/8.
• Adding indices when the bases are different, such as 2³ × 3² = 6⁵. Fix: Add indices only for the same base. Here 2³ × 3² = 8 × 9 = 72.
• Writing (aᵐ)ⁿ = a^(m+n). Fix: A power raised to a power multiplies: (2³)² = 2⁶ = 64, not 2⁵.
• Writing log(m + n) = log m + log n. 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). Fix: log(m ÷ n) = log m − log n. A ratio of two logs is the change of base form: log m ÷ log n = logₙm.
• Writing log (m + n) = log m + log n. 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). Fix: log (m ÷ n) = log m − log n. A ratio of two logs is a change of base, logₙ m.

Exam tips

• Most questions are about converting to a common prime base. Memorise squares to 15 and cubes to 10, plus powers of 2, 3 and 5.
• Read the sign of each index carefully. Options are often built from the common sign errors.
• For fractional indices, take the root first. 64^(2/3) is easier as 4² than as the cube root of 4096.
• Check whether the question says a ≠ 0 or gives a condition before using a⁰ = 1.
• With 50 questions in one hour, spend under a minute here. If a question has unfamiliar bases, try substituting a simple value.
• Memorise squares and cubes up to 15 and powers of 2, 3 and 5. They make prime-base conversion instant.
• Questions often hide the same base, such as 4, 8, 16 for base 2 or 9, 27, 81 for base 3. Spot the family first.
• If the algebra gets messy, test the four options. With no negative marking, a quick substitution is safe and fast.