Year 10 Maths

Logarithms

Understand logarithms as the inverse of exponentiation, apply log laws, solve exponential equations, and use the change of base rule.

What Is a Logarithm?

A logarithm answers the question: “What power do I raise the base to in order to get this number?” It is the inverse of exponentiation.

Definition

If a^x = b, then log_a(b) = x

Read as: “log base a of b equals x”

2³ = 8

log₂(8) = 3

10² = 100

log₁₀(100) = 2

5⁰ = 1

log₅(1) = 0

Just as index laws simplify expressions with powers, log laws simplify expressions involving logarithms.

Product Law

log_a(mn) = log_a(m) + log_a(n)

Quotient Law

log_a(m/n) = log_a(m) − log_a(n)

Power Law

log_a(m^p) = p × log_a(m)

Special Values

log_a(1) = 0 • log_a(a) = 1

To solve equations where the variable is in the exponent, take logs of both sides.

Step 1: Take log₁₀ of both sides: log(3^x) = log(20)

Step 2: Apply power law: x × log(3) = log(20)

Step 3: Solve: x = log(20) / log(3) = 1.301 / 0.477 ≈ 2.727

Change of Base Rule

log_a(b) = log_c(b) log_c(a)

Use this to evaluate logs with any base on a calculator (which typically has log₁₀ and ln).

Using change of base: log₅(40) = log(40) / log(5) = 1.602 / 0.699 ≈ 2.292

●A logarithm is the inverse of exponentiation: if a^x = b then log_a(b) = x.
●Product law: log(mn) = log(m) + log(n). Quotient law: log(m/n) = log(m) − log(n).
●Power law: log(m^p) = p log(m). Use this to “bring down” exponents.
●To solve exponential equations, take logs of both sides and apply the power law.
●Change of base: log_a(b) = log(b) / log(a) lets you evaluate any log on a calculator.