Logarithms

Definition of a logarithm, common and natural logs, the laws of logarithms, change of base, characteristic and mantissa, antilog, solving log and exponential equations, and quick worked examples.

Logarithms

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📑 Contents (6 sections)

Last reviewed 4 Nov 2026 · 3 min read

1. Definition

DefinitionLogarithm

If $a^{x} = N$ (with $a > 0$ , $a \neq = 1$ , $N > 0$ ), then $x = lo g_{a} N$ — "the log of $N$ to the base $a$ ".

Exponential form	Log form
$2^{5} = 32$	$lo g_{2} 32 = 5$
$1 0^{3} = 1000$	$lo g_{10} 1000 = 3$
$5^{- 2} = 1/25$	$lo g_{5} (1/25) = - 2$

Common log: base 10, written $lo g N$ . Natural log: base $e \approx 2.718$ , written $ln N$ .
The log of zero or a negative number is not defined (for real numbers).

2. Basic values

lo g_{a} 1 = 0, lo g_{a} a = 1, a^{l o g_{a} N} = N

Common logs to remember	Value
$lo g 2$	0.3010
$lo g 3$	0.4771
$lo g 5$	0.6990
$lo g 7$	0.8451

3. Laws of logarithms

FormulaLaws

Product: $lo g_{a} (M N) = lo g_{a} M + lo g_{a} N$
Quotient: $lo g_{a} (M / N) = lo g_{a} M - lo g_{a} N$
Power: $lo g_{a} (M^{p}) = p lo g_{a} M$
Change of base: $lo g_{a} N = \frac{lo g _{b} N}{lo g _{b} a}$
Reciprocal: $lo g_{a} b = \frac{1}{lo g _{b} a}$

Common MistakeCommon error

$lo g (M + N) \neq = lo g M + lo g N$ . The product rule applies to a product, not a sum.

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