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Graphing Logarithmic Functions

The function $y={\log }_{b}x$ is the inverse function of the exponential function $y=b^x$ .

Consider the function $y=3^x$ . It can be graphed as:

The graph of inverse function of any function is the reflection of the graph of the function about the line $y=x$ . So, the graph of the logarithmic function $y={\log }_3\left(x\right)$ which is the inverse of the function $y=3^x$ is the reflection of the above graph about the line $y=x$ .

$\begin{array}{|cccccccc|}\hline x& \frac{1}{9}& \frac{1}{3}& 1& 3& 9& 27& 81\\ y={\log }_{3}x& -2& -1& 0& 1& 2& 3& 4\\ \hline\end{array}$

The domain of the function is the set of all positive real numbers.

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

$\begin{array}{|cccccccc|}\hline x& \frac{1}{1000}& \frac{1}{100}& \frac{1}{10}& 1& 10& 100& 1000\\ y=\log x& -3& -2& -1& 0& 1& 2& 3\\ \hline\end{array}$

The logarithmic function, $y={\log }_b\left(x\right)$ , can be shifted $k$ units vertically and $h$ units horizontally with the equation $y={\log }_b(x+h)+k$ .

Vertical shift

If  $k>0$ , the graph would be shifted upwards.

If  $k<0$ , the graph would be shifted  downwards.

Horizontal Shift

If  $h>0$ , the graph would be shifted  left.

If  $h<0$ , the graph would be shifted  right.

Consider the logarithmic function $y=\left[{\log }_2(x+1)-3\right]$ . This can be obtained by translating the parent graph $y={\log }_2\left(x\right)$ a couple of times.

Consider the graph of the function $y={\log }_2\left(x\right)$ .

Since $h=1$ , $y=\left[{\log }_2(x+1)\right]$ is the translation of $y={\log }_2\left(x\right)$ by one unit to the left.

Now, $k=-3$ . The graph of $y=\left[{\log }_2(x+1)\right]$ will be shifted $3$ units down to get $y=\left[{\log }_2(x+1)\right]-3$ .

You may recall that logarithmic functions are defined only for positive real numbers. This is because, for negative values, the associated exponential equation has no solution. For example, $3^x=-1$ has no real solution, so ${\log }_3(-1)$ is undefined.

So, what about a function like $y={\log }_4(-x)$ ?

This is defined only for negative values of $x$ .

Find the values of the function for a few negative values of $x$ . For an easier calculation you can use the exponential form of the equation, $4^y=-x$ .

$\begin{array}{|ccccccc|}\hline x& -1& -2& -4& -8& -16& -32\\ \begin{array}{l}y={\log }_4(-x)\\ \text{or}\\ 4^y=-x\end{array}& 0& \frac{1}{2}& 1& 1\frac{1}{2}& 2& 2\frac{1}{2}\\ \hline\end{array}$

Plot the points and join them by a smooth curve.

You can see that the graph is the reflection of the graph of the function $y={\log }_4\left(x\right)$ about the $y$ -axis.