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











Sketch a graph of $\text{\hspace{0.17em}}f(x)=\mathrm{log}(x)\text{\hspace{0.17em}}$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.
Before graphing $\text{\hspace{0.17em}}f(x)=\mathrm{log}(x),$ identify the behavior and key points for the graph.
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 $\text{\hspace{0.17em}}f(x)=\mathrm{log}(x).\text{\hspace{0.17em}}$ 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.
Solve $\text{\hspace{0.17em}}4\mathrm{ln}\left(x\right)+1=2\mathrm{ln}\left(x1\right)\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}2\mathrm{ln}\left(x1\right)\text{\hspace{0.17em}}$ 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, $\text{\hspace{0.17em}}x\approx \mathrm{1.339.}$
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.
$x\approx 3.049$
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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