# Step By Step Calculus » 7.3 - Logarithmic and Inverse Hyperbolic Functions

Synopsis
The exponential function a^xa^x is strictly decreasing if a\in(0,1)a\in(0,1) and strictly increasing if a>1a>1. The inverse of a^{x}a^{x} is called the logarithm with base aa, written \log_{a}(x)\log_{a}(x), which is also strictly decreasing if a\in(0,1)a\in(0,1) and strictly increasing if a>1a>1. For the canonical base ee, the inverse of the exponential e^{x}e^{x} is written \ln(x)\ln(x) and called the natural logarithm. The logarithmic functions have special properties which are a direct result of their definition as the inverses of exponential functions. If a\in (0,1)\cup (1,\infty )a\in (0,1)\cup (1,\infty ), with xx and y y positive and r\in \mathbb{ R}r\in \mathbb{ R}, then
(Log multiplication)\displaystyle \log_{a}(xy)=\log _{a}(x)+\log _{a}(y)\displaystyle \log_{a}(xy)=\log _{a}(x)+\log _{a}(y).
(Log division)\displaystyle \log_{a}\left(\frac{x}{y}\right)=\log _{a}(x)-\log _{a}(y)\displaystyle \log_{a}\left(\frac{x}{y}\right)=\log _{a}(x)-\log _{a}(y).
(Log exponent)\displaystyle \log_{a}(x^{r})=r\log _{a}(x)\displaystyle \log_{a}(x^{r})=r\log _{a}(x).
(Log base change)\displaystyle \log_a(x)=\frac{\log_b(x)}{\log_b(a)}\displaystyle \log_a(x)=\frac{\log_b(x)}{\log_b(a)}.
As a consequence of Log base change, we get a change of base formula for exponentials.
\displaystyle a^x=b^{\log_b (a^x)}=b^{x\log_b(a)}.
\displaystyle a^x=b^{\log_b (a^x)}=b^{x\log_b(a)}.
We derive formulas for the inverses of the hyperbolic functions using the formula for finding the inverse of a composite function: