# Exponential and Logarithmic Functions

### Exponential function

If $b$ is any fixed number such that $b>0$ and $b\ne 1$ then an exponential function is a function of the form $$f(x)=b^x$$ where $b$ is called the base and $x$ can be any real number.

Fig 1: Graph of exponential function

The graph of $y=2^x$ is shown in Fig 1. The following common properties of exponential functions can seen from the graph:

1. The graph of $b^x$ will always contain the point $(0,1)$.
2. For every possible $b$ we have $b^x>0$.
3. If $b > 1$ then the graph of $b^x$ will increase as $x$ increases (note that if $0 < b <1$, the graph of $b^x$ will decrease as $x$ increases).
4. If $b^x = b^y$, then $x=y$.

An important base used in exponential functions is the base $e$ which is defined by $$e=\lim_{x\to\infty}\left(1+\frac1x\right)^x\approx 2.718281828.$$

$e$ is an irrational number which has important properties in many areas of engineering.

### Logarithmic function

If $b$ is any fixed number such that $b>0$ and $b \ne 1$, and $x > 0$ is any positive real number, then a logarithmic function (with base $b$) is a function of the form $$y=\log_b x$$ which is equivalent to the exponential form $b^y=x$.

The inverse of an exponential function is a logarithmic function. The inverse of a function can be thought of as the function obtained by exchanging the $x$ and $y$ coordinates. This reflects the graph about the line $y=x$. As seen from the graph in Fig 2, the logarithmic curve (blue) is the reflection of the exponential curve (red).

Fig 2: Graph of exponential and logarithmic functions

### Comparison of exponential and logarithmic functions

PropertyExponentialLogarithmic
Function $y=b^x,\;b>0,\;b\ne 1$ $y=\log_b x,\;b>0,\;b\ne 1$
Domain All real numbers $x>0$
Range $y>0$ All real numbers
Intercept $x=0'\;y=1$ $x=1\;y=0$
Increasing When $b>1$ When $b>1$
Decreasing When $0<b<1$ When $0<b<1$