Exponential functions and logarithms

An exponential function is a function where the variable is in the exponent, usually written as $f(x) = a^x$ for some constant $a > 0$, $a \neq 1$. A logarithm is the inverse operation to exponentiation: $\log_a b$ is the exponent to which $a$ must be raised to get $b$.

Important properties

• For $a > 0$, $a \neq 1$, the function $f(x) = a^x$ is always positive.

• Exponential functions grow (or decay) much faster than polynomial functions.

• The logarithm $\log_a b$ answers the question: 'To what power must $a$ be raised to get $b$?'

• Exponential and logarithmic functions are inverses: $a^{\log_a x} = x$ and $\log_a(a^x) = x$.

• Laws of exponents: $a^{x+y} = a^x a^y$, $a^{x-y} = a^x / a^y$, $(a^x)^y = a^{xy}$.

• Laws of logarithms: $\log_a(xy) = \log_a x + \log_a y$, $\log_a(x/y) = \log_a x - \log_a y$, $\log_a(x^k) = k \log_a x$.