8.7: Graphing Logarithmic Functions
Your math homework assignment is to find out which quadrants the graph of the function \begin{align*}f(x)= 4\ln(x + 3)\end{align*}
Guidance
Now that we are more comfortable with using these functions as inverses, let’s use this idea to graph a logarithmic function. Recall that functions are inverses of each other when they are mirror images over the line \begin{align*}y=x\end{align*}
Recall that an exponential function has a horizontal asymptote. Because the logarithm is its inverse, it will have a vertical asymptote. The general form of a logarithmic function is \begin{align*}f(x)=\log_b(xh)+k\end{align*}
Example A
Graph \begin{align*}y=\log_3(x4)\end{align*}
Solution:
To graph a logarithmic function without a calculator, start by drawing the vertical asymptote, at \begin{align*}x=4\end{align*}
The domain is \begin{align*}x>4\end{align*}
Example B
Is (16, 1) on \begin{align*}y=\log (x6)\end{align*}
Solution: Plug in the point to the equation to see if it holds true.
\begin{align*}1 &= \log(166) \\
1 &= \log 10 \\
1 &= 1\end{align*}
Yes, this is true, so (16, 1) is on the graph.
Example C
Graph \begin{align*}f(x)=2 \ln(x+1)\end{align*}
Solution: To graph a natural log, we need to use a graphing calculator. Press \begin{align*}Y=\end{align*}
Intro Problem Revisit the vertical asymptote of the function \begin{align*}f(x)= 4\ln(x + 3)\end{align*}
Guided Practice
1. Graph \begin{align*}y=\log_{\frac{1}{4}} x+2\end{align*}
2. Graph \begin{align*}y=\log x\end{align*}
3. Is (2, 1) on the graph of \begin{align*}f(x)=\log_{\frac{1}{2}} (x+4)\end{align*}
Answers
1. First, there is a vertical asymptote at \begin{align*}x=0\end{align*}
To graph a logarithmic function using a TI83/84, enter the function into \begin{align*}Y=\end{align*}
\begin{align*}Y= \frac{\log(x)}{\log \left ( \frac{1}{4} \right )}+2\end{align*}
To see a table of values, press \begin{align*}2^{nd} \rightarrow \end{align*}
2. The keystrokes are \begin{align*}Y=\log(x)\end{align*}
The domain is \begin{align*}x>0\end{align*}
3. Plug (2, 1) into \begin{align*}f(x)=\log_{\frac{1}{2}} (x+4)\end{align*}
\begin{align*}1 &= \log_{\frac{1}{2}} (2+4) \\
1 &= \log_{\frac{1}{2}} 2 \rightarrow \frac{1}{2}^x=2 \\
1 & \ne 1 \end{align*}
Therefore, (2, 1) is not on the graph. However, (2, 1) is.
Practice
Graph the following logarithmic functions without using a calculator. State the equation of the asymptote, the domain and the range of each function.

\begin{align*}y=\log_5 x\end{align*}
y=log5x 
\begin{align*}y=\log_2(x+1)\end{align*}
y=log2(x+1) 
\begin{align*}y=\log(x)4\end{align*}
y=log(x)−4 
\begin{align*}y=\log_{\frac{1}{3}}(x1)+3\end{align*}
y=log13(x−1)+3 
\begin{align*}y=\log_{\frac{1}{2}}(x+3)5\end{align*}
y=−log12(x+3)−5 
\begin{align*}y=\log_4(2x)+2\end{align*}
y=log4(2−x)+2
Graph the following logarithmic functions using your graphing calculator.

\begin{align*}y=\ln (x+6)1\end{align*}
y=ln(x+6)−1 
\begin{align*}y=\ln (x1)+2\end{align*}
y=−ln(x−1)+2 
\begin{align*}y=\log(1x)+3\end{align*}
y=log(1−x)+3 
\begin{align*}y=\log(x+2)4\end{align*}
y=log(x+2)−4  How would you graph \begin{align*}y=\log_4 x\end{align*}
y=log4x on the graphing calculator? Graph the function.  Graph \begin{align*}y=\log_{\frac{3}{4}}x\end{align*}
y=log34x on the graphing calculator.  Is (3, 8) on the graph of \begin{align*}y=\log_3 (2x3)+7\end{align*}
y=log3(2x−3)+7 ?  Is (9, 2) on the graph of \begin{align*}y=\log_{\frac{1}{4}} (x5)\end{align*}
y=log14(x−5) ?  Is (4, 5) on the graph of \begin{align*}y=5 \log_2 (8x)\end{align*}
y=5log2(8−x) ?
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Asymptotes
An asymptote is a line on the graph of a function representing a value toward which the function may approach, but does not reach (with certain exceptions).operation
Operations are actions performed on variables, constants, or expressions. Common operations are addition, subtraction, multiplication, and division.Image Attributions
Here you'll learn how to graph a logarithmic function by hand and using a calculator.