Logarithms

Logarithms are the inverse of indices. In this topic you will learn what a logarithm means, the basic laws of logs, how to solve simple equations, and how to change base. We use base 10 and base e (natural logs) most often.

  • Definition and link to indices
  • Laws of logarithms
  • Solving equations
  • Change of base

Lessons for Logarithms

What is a Logarithm?

DefinitionIf ax = b, then logab = x.We say: log base a of b equals x. (a is the base, …

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Laws of Logarithms

The three main laws (same base)1. Multiplication: loga(xy) = logax + logay2. Division: loga(x/y) = logax - logay3. Power: loga(xn) …

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Solving Equations with Logs

Type 1: logax = bConvert to index form: x = abExample: log2x = 5   ⇒   x = 25 …

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Change of Base

Formulalogab = (logcb) / (logca)Useful when your calculator only has log10 (log) or ln (loge).Exampleslog210 = log 10 / log …

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Exercises for Logarithms

Convert between index and log form

Write in log form:34 = 81103 = 100071 = 7Write in index form:log232 = 5log100.01 = -2log464 = 3

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Simplify using log laws

Simplify as a single logarithm:log 3 + log 7log520 - log542 log 4log212 - log233 log102 + log105

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Solve for x

Solve (base 10 unless shown):log x = 2log3x = 4log x + log 5 = 1log2x - log23 = 22 …

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Use change of base

Evaluate to 3 s.f. using log10:log27log315log50.2log432

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