A6: Exponential and logarithmic functions

For $a>0$ and $a\ne1$, $y=a^x$ is positive and has horizontal asymptote $y=0$. The function $y=\log_a x$ is its inverse, so their graphs reflect in $y=x$. The natural exponential and logarithm use base $e$.

Logarithm laws convert multiplication into addition, division into subtraction, and powers into multipliers. Change of base allows a logarithm to be evaluated using another available base.

Rewrite exponential terms with a common base where possible. Otherwise take logarithms after isolating the exponential expression. For logarithmic equations, combine valid logarithms, solve the resulting algebraic equation, and reject any value that makes an original argument non-positive.

A repeated proportional increase or decrease can be represented by $y=Ab^x$ or $y=Ae^{kx}$. Define the initial value and growth factor, use only meaningful inputs, and interpret predictions within the model's practical range.