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Given a logarithmic function, identify the domain.
What is the domain of $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{2}(x+3)?$
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 $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{2}(x+3)\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\left(-3,\infty \right).$
What is the domain of $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{5}(x-2)+1?$
$\left(2,\infty \right)$
What is the domain of $\text{\hspace{0.17em}}f(x)=\mathrm{log}(5-2x)?$
The logarithmic function is defined only when the input is positive, so this function is defined when $\text{\hspace{0.17em}}5\u20132x>0.\text{\hspace{0.17em}}$ Solving this inequality,
The domain of $\text{\hspace{0.17em}}f(x)=\mathrm{log}(5-2x)\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\left(\u2013\infty ,\frac{5}{2}\right).$
What is the domain of $\text{\hspace{0.17em}}f(x)=\mathrm{log}(x-5)+2?$
$\left(5,\infty \right)$
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 $\text{\hspace{0.17em}}x={\mathrm{log}}_{2}(y)\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}f(x)={2}^{x}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g(x)={\mathrm{log}}_{2}(x).\text{\hspace{0.17em}}$ See [link] .
$f(x)={2}^{x}$ | $\left(-3,\frac{1}{8}\right)$ | $\left(-2,\frac{1}{4}\right)$ | $\left(-1,\frac{1}{2}\right)$ | $\left(0,1\right)$ | $\left(1,2\right)$ | $\left(2,4\right)$ | $\left(3,8\right)$ |
$g(x)={\mathrm{log}}_{2}\left(x\right)$ | $\left(\frac{1}{8},-3\right)$ | $\left(\frac{1}{4},-2\right)$ | $\left(\frac{1}{2},-1\right)$ | $\left(1,0\right)$ | $\left(2,1\right)$ | $\left(4,2\right)$ | $\left(8,3\right)$ |
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:
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 $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{b}\left(x\right):$
See [link] .
[link] shows how changing the base $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ in $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ 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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