Exponential and Logarithmic Functions

Exponential functions have the form $f(x) = a^x$ (with $a > 0$, $a \neq 1$), or more usefully $f(x) = k\,a^{x} + c$. The variable sits in the exponent, which produces very rapid growth or decay. For a basic exponential $y = a^x$: the graph passes through $(0, 1)$ because $a^0 = 1$; it is always positive; and it has a horizontal asymptote at $y = 0$. When $a > 1$ the curve rises (growth); when $0 < a < 1$ it falls (decay). A particularly important base is Euler's number $e \approx 2.718$, giving the natural exponential $y = e^x$.

Logarithmic functions are the inverses of exponentials. The statement $y = \log_a x$ means exactly the same as $a^y = x$. Because logs and exponentials are inverse operations, their graphs are reflections of each other in the line $y = x$. The graph of $y = \log_a