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14.4PureCore

Logarithms

A logarithm answers the question “what power do I raise the base to?” It is the inverse of an exponential, and it is the tool for getting a variable down out of an exponent.

25 min Exponentials and Logarithms
Edexcel AS Level Maths: 14.4 LogarithmsCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Understand a logarithm as the inverse of a power
  • Convert between exponential and log form
  • Evaluate simple logarithms
  • Use the key special values
1

Logarithm = inverse of a power

The statement log⁡ab=c\log_a b = c means “aa to the power cc equals bb”. So a logarithm is just a rearranged power.

log⁡ab=c  ⟺  ac=b\log_a b = c \;\Longleftrightarrow\; a^{c} = b
Log form and exponential form say the same thing.
Example 1Evaluate log⁡28\log_2 8.
1Ask: 2 to what power is 8? 23=82^3 = 8.
Answer33
2

Converting between forms

Being able to flip between log⁡ab=c\log_a b = c and ac=ba^c = b is the core skill — it is how you solve equations with the unknown in the power.

Example 1Write 103=100010^3 = 1000 in log form.
1Base 10, power 3, result 1000.
Answerlog⁡101000=3\log_{10} 1000 = 3

Tip — Read log_a b as “the power that turns a into b”. That phrasing makes evaluations instant.

3

Special values

A few results follow straight from the definition: log⁡aa=1\log_a a = 1 (since a1=aa^1 = a) and log⁡a1=0\log_a 1 = 0 (since a0=1a^0 = 1).

log⁡aa=1,log⁡a1=0\log_a a = 1, \qquad \log_a 1 = 0
Two values worth knowing instantly.

Formula recap

log⁡ab=c⇔ac=b\log_a b = c \Leftrightarrow a^{c} = b
Definition / form conversion.
log⁡aa=1\log_a a = 1
Log of the base is 1.
log⁡a1=0\log_a 1 = 0
Log of 1 is 0.

Common mistakes to avoid

Confusing which number is the base.
In log_a b, a is the base — the same a that gets raised to a power.
Thinking log_a 1 = 1.
log_a 1 = 0, because a⁰ = 1.

Key takeaways

  • log_a b = c means a^c = b — a log is the inverse of a power.
  • Convert freely between log and exponential form.
  • log_a a = 1 and log_a 1 = 0.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 8 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.