# 6.4 Graphs of logarithmic functions  (Page 6/8)

 Page 6 / 8

Given a logarithmic function with the parent function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right),$ graph a translation.

1. Draw the vertical asymptote, $\text{\hspace{0.17em}}x=0.$
1. Draw the vertical asymptote, $\text{\hspace{0.17em}}x=0.$
1. Plot the x- intercept, $\text{\hspace{0.17em}}\left(1,0\right).$
1. Plot the x- intercept, $\text{\hspace{0.17em}}\left(1,0\right).$
1. Reflect the graph of the parent function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ about the x -axis.
1. Reflect the graph of the parent function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ about the y -axis.
1. Draw a smooth curve through the points.
1. Draw a smooth curve through the points.
1. State the domain, $\text{\hspace{0.17em}}\left(0,\infty \right),$ the range, $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote $\text{\hspace{0.17em}}x=0.$
1. State the domain, $\text{\hspace{0.17em}}\left(-\infty ,0\right),$ the range, $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote $\text{\hspace{0.17em}}x=0.$

## Graphing a reflection of a logarithmic function

Sketch a graph of $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{log}\left(-x\right)\text{\hspace{0.17em}}$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Before graphing $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{log}\left(-x\right),$ identify the behavior and key points for the graph.

• Since $\text{\hspace{0.17em}}b=10\text{\hspace{0.17em}}$ is greater than one, we know that the parent function is increasing. Since the input value is multiplied by $\text{\hspace{0.17em}}-1,$ $f\text{\hspace{0.17em}}$ is a reflection of the parent graph about the y- axis. Thus, $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{log}\left(-x\right)\text{\hspace{0.17em}}$ will be decreasing as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ moves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptote $\text{\hspace{0.17em}}x=0.\text{\hspace{0.17em}}$
• The x -intercept is $\text{\hspace{0.17em}}\left(-1,0\right).$
• We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.

The domain is $\text{\hspace{0.17em}}\left(-\infty ,0\right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=0.$

Graph $\text{\hspace{0.17em}}f\left(x\right)=-\mathrm{log}\left(-x\right).\text{\hspace{0.17em}}$ State the domain, range, and asymptote.

The domain is $\text{\hspace{0.17em}}\left(-\infty ,0\right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=0.$

Given a logarithmic equation, use a graphing calculator to approximate solutions.

1. Press [Y=] . Enter the given logarithm equation or equations as Y 1 = and, if needed, Y 2 = .
2. Press [GRAPH] to observe the graphs of the curves and use [WINDOW] to find an appropriate view of the graphs, including their point(s) of intersection.
3. To find the value of $\text{\hspace{0.17em}}x,$ we compute the point of intersection. Press [2ND] then [CALC] . Select “intersect” and press [ENTER] three times. The point of intersection gives the value of $\text{\hspace{0.17em}}x,$ for the point(s) of intersection.

## Approximating the solution of a logarithmic equation

Solve $\text{\hspace{0.17em}}4\mathrm{ln}\left(x\right)+1=-2\mathrm{ln}\left(x-1\right)\text{\hspace{0.17em}}$ graphically. Round to the nearest thousandth.

Press [Y=] and enter $\text{\hspace{0.17em}}4\mathrm{ln}\left(x\right)+1\text{\hspace{0.17em}}$ next to Y 1 =. Then enter $\text{\hspace{0.17em}}-2\mathrm{ln}\left(x-1\right)\text{\hspace{0.17em}}$ next to Y 2 = . For a window, use the values 0 to 5 for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and –10 to 10 for $\text{\hspace{0.17em}}y.\text{\hspace{0.17em}}$ Press [GRAPH] . The graphs should intersect somewhere a little to right of $\text{\hspace{0.17em}}x=1.$

For a better approximation, press [2ND] then [CALC] . Select [5: intersect] and press [ENTER] three times. The x -coordinate of the point of intersection is displayed as 1.3385297. (Your answer may be different if you use a different window or use a different value for Guess? ) So, to the nearest thousandth, $\text{\hspace{0.17em}}x\approx 1.339.$

Solve $\text{\hspace{0.17em}}5\mathrm{log}\left(x+2\right)=4-\mathrm{log}\left(x\right)\text{\hspace{0.17em}}$ graphically. Round to the nearest thousandth.

$x\approx 3.049$

## Summarizing translations of the logarithmic function

Now that we have worked with each type of translation for the logarithmic function, we can summarize each in [link] to arrive at the general equation for translating exponential functions.

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