Hard

Using the Change of Base Formula

Simple Explanation

Most calculators only compute logarithms in base 10 or base e. The change of base formula, log_b M = log_a M / log_a b, converts a logarithm in any base b into a ratio of logarithms in a base a your calculator understands.

Why Do We Need It?

Without this formula, evaluating something like log₇(50) would be impossible on an ordinary calculator — change of base makes every logarithm computable, regardless of its base.

Formula

Change of Base Formula

log_b M = log_a M / log_a b

Converts a logarithm in one base (b) into a ratio of logarithms in a different, more convenient base (a) — usually base 10 or base e, so a calculator can evaluate it.

b: — the original base of the logarithm
a: — the new base being switched to (commonly 10 or e)
M: — the argument of the logarithm

When to use it: Whenever you need to evaluate a logarithm in a base your calculator does not have a direct button for (any base other than 10 or e).

Worked Example

Evaluate a logarithm using change of base

Evaluate log₇(50) using base-10 logarithms (log₁₀50 ≈ 1.699, log₁₀7 ≈ 0.845).

Why Does This Work?

Let log_b M = y, so bʸ = M. Taking log_a of both sides gives log_a(bʸ) = log_a M, and the power rule turns the left side into y · log_a b — solving for y reproduces exactly the change of base formula.

Real-Life Example

Comparing growth rates with different bases

Two competing investment plans are modeled with different exponential bases, and an analyst needs to compare their logarithms directly.

Change of base lets the analyst convert both to the same base (say, base 10) so their logarithmic values can be directly compared using an ordinary calculator.

Practice

Using log₁₀2 ≈ 0.301 and log₁₀3 ≈ 0.477, evaluate log₃(2) to 2 decimal places.

Hard

Common mistake

Writing the formula upside down (log_a b / log_a M instead of log_a M / log_a b) — the original argument M always goes in the numerator, and the original base b always goes in the denominator.

Quick Review

log_b M = log_a M / log_a b, for any convenient new base a.
Almost always used with a = 10 or a = e, to match a calculator.
The original argument goes on top; the original base goes on the bottom.