# Graphing Exponential and Logarithmic Functions

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Graphing Exponential Functions
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A simple exponential function to graph is $y={2}^{x}$ .

$x$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ |

$y={2}^{x}$ | $\frac{1}{8}$ | $\frac{1}{4}$ | $\frac{1}{2}$ | $1$ | $2$ | $4$ | $8$ |

Notice that the graph has the $x$ -axis as an asymptote on the left, and increases very fast on the right.

Changing the base changes the shape of the graph.

Replacing $x$ with $-x$ reflects the graph across the $y$ -axis; replacing $y$ with $-y$ reflects it across the $x$ -axis.

Replacing $x$ with $x+h$ translates the graph $h$ units to the left.

Replacing $y$ with $y-k$ (which is the same as adding $k$ to the right side) translates the graph $k$ units up.

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Graphing Logarithmic Functions
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The function $y={\mathrm{log}}_{b}x$ is the inverse function of $y={b}^{x}$ . So, it is the reflection of that graph across the diagonal line $y=x$ .

When no base is written, assume that the log is base $10$ .

$x$ | $\frac{1}{1000}$ | $\frac{1}{100}$ | $\frac{1}{10}$ | $1$ | $10$ | $100$ | $1000$ |

$y=\mathrm{log}x$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ |