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 Video by Zeeshan Zamurred Exponentials and Logarithms

Edexcel AS Level Maths: 14.4 LogarithmsWatch the full walkthrough 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

Logarithm = inverse of a power

The statement $lo g_{a} b = c$ means “ $a$ to the power $c$ equals $b$ ”. So a logarithm is just a rearranged power.

lo g_{a} b = c ⟺ a^{c} = b

Log form and exponential form say the same thing.

1Ask: 2 to what power is 8?

2^{3} = 8

Answer

3

Converting between forms

Being able to flip between $lo g_{a} b = c$ and $a^{c} = b$ is the core skill — it is how you solve equations with the unknown in the power.

1Base 10, power 3, result 1000.

Answer

lo g_{10} 1000 = 3

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

Special values

A few results follow straight from the definition: $lo g_{a} a = 1$ (since $a^{1} = a$ ) and $lo g_{a} 1 = 0$ (since $a^{0} = 1$ ).

lo g_{a} a = 1, lo g_{a} 1 = 0

Two values worth knowing instantly.

Formula recap

lo g_{a} b = c \Leftrightarrow a^{c} = b

Definition / form conversion.

lo g_{a} a = 1

Log of the base is 1.

lo g_{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.

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