Exponents & logarithms cheat sheet

Logarithms are exponents in disguise, so every log rule mirrors an exponent rule. Learn the pairs together and half the list comes free.

Practise algebra

Exponent laws

$a^{m} a^{n} = a^{m + n}$
$\frac{a ^{m}}{a ^{n}} = a^{m - n}$
$(a^{m})^{n} = a^{mn}$
$a^{- n} = \frac{1}{a ^{n}}$
$a^{0} = 1 (a \neq = 0)$
$a^{m / n} = n a^{m}$

Radicals

$ab = a b (a, b \geq 0)$
$\frac{a}{b} = \frac{a}{b}$
$\frac{1}{a} = \frac{a}{a}$ Rationalise
$a^{2} = ∣ a ∣$

Logarithm rules

$lo g_{b} (x y) = lo g_{b} x + lo g_{b} y$
$lo g_{b} (\frac{x}{y}) = lo g_{b} x - lo g_{b} y$
$lo g_{b} (x^{n}) = n lo g_{b} x$
$lo g_{b} b = 1, lo g_{b} 1 = 0$
$b^{l o g_{b} x} = x, lo g_{b} (b^{x}) = x$
$lo g_{b} x = \frac{l n x}{l n b}$ Change of base

Solving

$b^{x} = c \Rightarrow x = lo g_{b} c$
$ln e^{x} = x, e^{l n x} = x$
$lo g x = lo g_{10} x$ Common log convention

Related calculators

Logarithm Calculator Exponent Calculator