## P.5 Exponential and Logarithmic Functions

### Exponential Functions

Exponential functions are when a constant is raised a power, $f(x)=2^x$. These have important properties that will come up often during the course.

\[\begin{align*}
a^0 = 1 \quad&& a^xa^y=a^{x+y} \quad&& \left(a^x\right)^y=a^{xy} \quad&& (ab)^x = a^xb^x \\[1em]
\frac{a^x}{a^y} = a^{x-y} \quad&& \left(\frac{a}{b}\right)^x = \frac{a^x}{b^x} \quad&& a^{-x}=\frac{1}{a^x}
\end{align*}\]

Absent of any transformations, the domain of exponential functions is all real numbers, the range is $(0,\infty)$, the $y$-intercept is $(0,1)$, and the function is one-to-one. Keep in mind that when $a>1$, you see exponential growth, whereas when $0<a<1$, you will a graph showing exponential decay.

### The Number $e$

$e$, sometime’s called Euler’s Number (pronounces *oiler*), will pop up a lot. Its importance is tough to explain to anyone outside of math. It’s working in the background with things like compound interest and population growth, and is just one of those numbers that seems to be describing a lot of the universe, like $\pi$.

For now, just make sure you know it exists, its irrational, its value is $\approx 2.71828$, and that is the solution to natural logarithms.

### The Natural Logarithm

Logarithms can have any base, but using a base of $e$ is the one that appears most common throughout math subjects. This one is inverse function of the natural exponential function, $e^x$

\[\ln x = b \quad \text{if and only if} \quad e^b = x\]

Being an inverse of an exponential function, the properties apply here, but of course all inverted. Domain and range are swapped, and the $y$-intercept is now an $x$-intercept.

There are also the algebraic properties.

\[\ln xy = \ln x + \ln y \qquad \ln \frac{x}{y} = \ln x - \ln y \qquad \ln x^z = z\ln x\]

The most common use of natural logarithms is solving exponential functions.

\[\begin{align*}
7 &= e^{x+1} \\
\ln 7 &= \ln\left(e^{x + 1}\right) && \text{Take natural log of both sides} \\
\ln 7 &= x+1 \\
x &= -1 + \ln 7
\end{align*}\]