4.4 Graphs of logarithmic functions  (Page 2/8)

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Given a logarithmic function, identify the domain.

1. Set up an inequality showing the argument greater than zero.
2. Solve for $\text{\hspace{0.17em}}x.$
3. Write the domain in interval notation.

Identifying the domain of a logarithmic shift

What is the domain of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{2}\left(x+3\right)?$

The logarithmic function is defined only when the input is positive, so this function is defined when $\text{\hspace{0.17em}}x+3>0.\text{\hspace{0.17em}}$ Solving this inequality,

The domain of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{2}\left(x+3\right)\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\left(-3,\infty \right).$

What is the domain of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{5}\left(x-2\right)+1?$

$\left(2,\infty \right)$

Identifying the domain of a logarithmic shift and reflection

What is the domain of $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{log}\left(5-2x\right)?$

The logarithmic function is defined only when the input is positive, so this function is defined when $\text{\hspace{0.17em}}5–2x>0.\text{\hspace{0.17em}}$ Solving this inequality,

The domain of $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{log}\left(5-2x\right)\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\left(–\infty ,\frac{5}{2}\right).$

What is the domain of $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{log}\left(x-5\right)+2?$

$\left(5,\infty \right)$

Graphing logarithmic functions

Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ along with all its transformations: shifts, stretches, compressions, and reflections.

We begin with the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right).\text{\hspace{0.17em}}$ Because every logarithmic function of this form is the inverse of an exponential function with the form $\text{\hspace{0.17em}}y={b}^{x},$ their graphs will be reflections of each other across the line $\text{\hspace{0.17em}}y=x.\text{\hspace{0.17em}}$ To illustrate this, we can observe the relationship between the input and output values of $\text{\hspace{0.17em}}y={2}^{x}\text{\hspace{0.17em}}$ and its equivalent $\text{\hspace{0.17em}}x={\mathrm{log}}_{2}\left(y\right)\text{\hspace{0.17em}}$ in [link] .

 $x$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$ ${2}^{x}=y$ $\frac{1}{8}$ $\frac{1}{4}$ $\frac{1}{2}$ $1$ $2$ $4$ $8$ ${\mathrm{log}}_{2}\left(y\right)=x$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$

Using the inputs and outputs from [link] , we can build another table to observe the relationship between points on the graphs of the inverse functions $\text{\hspace{0.17em}}f\left(x\right)={2}^{x}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)={\mathrm{log}}_{2}\left(x\right).\text{\hspace{0.17em}}$ See [link] .

 $f\left(x\right)={2}^{x}$ $\left(-3,\frac{1}{8}\right)$ $\left(-2,\frac{1}{4}\right)$ $\left(-1,\frac{1}{2}\right)$ $\left(0,1\right)$ $\left(1,2\right)$ $\left(2,4\right)$ $\left(3,8\right)$ $g\left(x\right)={\mathrm{log}}_{2}\left(x\right)$ $\left(\frac{1}{8},-3\right)$ $\left(\frac{1}{4},-2\right)$ $\left(\frac{1}{2},-1\right)$ $\left(1,0\right)$ $\left(2,1\right)$ $\left(4,2\right)$ $\left(8,3\right)$

As we’d expect, the x - and y -coordinates are reversed for the inverse functions. [link] shows the graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g.$

Observe the following from the graph:

• $f\left(x\right)={2}^{x}\text{\hspace{0.17em}}$ has a y -intercept at $\text{\hspace{0.17em}}\left(0,1\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)={\mathrm{log}}_{2}\left(x\right)\text{\hspace{0.17em}}$ has an x - intercept at $\text{\hspace{0.17em}}\left(1,0\right).$
• The domain of $\text{\hspace{0.17em}}f\left(x\right)={2}^{x},$ $\left(-\infty ,\infty \right),$ is the same as the range of $\text{\hspace{0.17em}}g\left(x\right)={\mathrm{log}}_{2}\left(x\right).$
• The range of $\text{\hspace{0.17em}}f\left(x\right)={2}^{x},$ $\left(0,\infty \right),$ is the same as the domain of $\text{\hspace{0.17em}}g\left(x\right)={\mathrm{log}}_{2}\left(x\right).$

Characteristics of the graph of the parent function, f ( x ) = log b ( x )

For any real number $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and constant $\text{\hspace{0.17em}}b>0,$ $b\ne 1,$ we can see the following characteristics in the graph of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right):$

• one-to-one function
• vertical asymptote: $\text{\hspace{0.17em}}x=0$
• domain: $\text{\hspace{0.17em}}\left(0,\infty \right)$
• range: $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)$
• x- intercept: $\text{\hspace{0.17em}}\left(1,0\right)\text{\hspace{0.17em}}$ and key point $\left(b,1\right)$
• y -intercept: none
• increasing if $\text{\hspace{0.17em}}b>1$
• decreasing if $\text{\hspace{0.17em}}0

[link] shows how changing the base $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ in $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. ( Note: recall that the function $\text{\hspace{0.17em}}\mathrm{ln}\left(x\right)\text{\hspace{0.17em}}$ has base $\text{\hspace{0.17em}}e\approx \text{2}.\text{718.)}$

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