Graphing Logarithmic Functions without Technology Solutions

• Because the graph of a    logarithmic    function, $y={\log }_{b}x$, is the inverse of an    exponential    function, $y=b^x$:
  • $\underline{\mathbf{ }\mathbf{ }\mathbf{ }\mathit{x}\mathbf{>}\mathbf{0}\mathbf{ }\mathbf{ }\mathbf{ }}$
  • The graph is reflected across $\underline{\mathbf{ }\mathbf{ }\mathbf{ }\mathit{y}\mathbf{=}\mathit{x}\mathbf{ }\mathbf{ }\mathbf{ }}$
  • The asymptote is the vertical line $\underline{\mathbf{ }\mathbf{ }\mathbf{ }\mathit{x}\mathbf{=}\mathbf{0}\mathbf{ }\mathbf{ }\mathbf{ }}$ 
  • There is an x-intercept, $\underline{ \mathbf{(}\mathbf{1}\mathbf{,}\mathbf{ }\mathbf{0}\mathbf{)} }$ but no y-intercept
  • The domain is: $\underline{ \mathbf{\left\{}\mathit{x}\mathbf{\right|}\mathit{x}\mathbf{\in }\mathbf{ℝ}\mathbf{,}\mathit{x}\mathbf{>}\mathbf{0}\mathbf{\}}\mathbf{ }\mathbf{or}\mathbf{ }\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{\infty }\mathbf{)} }$
  • The range is: $\underline{ \mathbf{\left\{}\mathit{y}\mathbf{\right|}\mathit{y}\mathbf{\in }\mathbf{ℝ}\mathbf{\}}\mathbf{ }\mathbf{or}\mathbf{ }\mathbf{(}\mathbf{–}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)} }$  
• A    sketch    of the exponential function as a dashed graph can help visualize its inverse, the logarithmic function.

• Because the value of most logarithms results in an    irrational    number, it is sometimes necessary to work with the    inverse    of the function.

• Then you can compare the    log    and the    exponential    functions.
• When graphing a logarithmic function without technology:

  •    Rewrite    it as its inverse.

  • Make a table of    rational values   .

  • Interchange the    x- and y-values    from the table.

  • Graph the    logarithm   .

Note

In the next few examples, you will graph simple logarithms from a table. In the next section, you will sketch more complicated graphs using technology.

Example 3

Compare f(x) and g(x).

$f\left(x\right)=2^x$           $g\left(x\right)={\log }_{2}x$

f(x) and g(x) are    inverses    because their    ordered pairs    are switched, and the graph of g(x) is    reflected over y = x  .

Example 4

Graph without technology. Describe the end behavior and the domain and range.

$y={\log }_{5}x$

Note

Even if all of the points in your table do not fit on the given coordinate plane, they can help you picture the general shape of the graph.

Explain

End behavior: As $x\to +\infty$, $\underline{ f\left(x\right)\to +\infty }$ and as $x\to 0$, $\underline{ f\left(x\right)\to –\infty }.$

Domain: $\underline{ \left\{x\right|x\in \mathrm{ℝ},x>0\} }$

Range: $\underline{ \left\{y\right|y\in \mathrm{ℝ}\} }$

Example 5

Graph the inverse of the given equation without technology. Describe the end behavior.

$q\left(x\right)={\left(\frac{5}{4}\right)}^{–x}$