Transformations of Graphs of Logarithmic Functions

Reflections of the Parent Function $y=log_b(x)$

The function $f(x)=−log_b(x)$

• reflects the parent function $y=log_b(x)$ about the $x$-axis.
• has domain, $(0, \infty)$, range, $(−\infty, \infty)$, and vertical asymptote, $x=0$, which are unchanged from the parent function.

The function $f(x)=log_b(−x)$

• reflects the parent function $y=log_b(x)$ about the $y$-axis.
• has domain $(−\infty, 0)$.
• has range, $(−\infty, \infty)$, and vertical asymptote, $x=0$, which are unchanged from the parent function.

HOW TO

Given a logarithmic function with the parent function $f(x)=log_b(x)$, graph a translation.

If $f(x)=−log_b(x)$	If $f(x)=log_b(-x)$
1. Draw the vertical asymptote, $x=0$.	1. Draw the vertical asymptote, $x=0$.
2. Plot the $x$-intercept, $(1,0)$.	2. Plot the $x$-intercept, $(1,0)$.
3. Reflect the graph of the parent function $f(x)=log_b(x)$ about the $x$-axis.	3. Reflect the graph of the parent function $f(x)=log_b(x)$ about the $y$-axis.
4. Draw a smooth curve through the points.	4. Draw a smooth curve through the points.
5. State the domain, $(0, \infty)$, the range, $(−\infty, \infty)$, and the vertical asymptote $x=0$.	5. State the domain, $(−\infty, 0)$ the range, $(−\infty, \infty)$ and the vertical asymptote $x=0$.

Table 3

Example 8

Graphing a Reflection of a Logarithmic Function

Sketch a graph of $f(x)=log(−x)$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Solution

Before graphing $f(x)=log(−x)$, identify the behavior and key points for the graph.

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

Figure 14

The domain is $(−\infty, 0)$, the range is $(−\infty, \infty)$, and the vertical asymptote is $x=0$.

Try It #8

Graph $f(x)=−log(−x)$.  State the domain, range, and asymptote.

HOW TO

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

1. Press [Y=]. Enter the given logarithm equation or equations as Y₁= and, if needed, Y₂=.
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 $x$, 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 $x$, for the point(s) of intersection.

Example 9

Approximating the Solution of a Logarithmic Equation

Solve $4 \ln(x)+1=−2ln(x−1)$ graphically. Round to the nearest thousandth.

Solution

Press [Y=] and enter $4 \ln(x)+1$ next to Y₁=. Then enter $−2 \ln(x−1)$ next to Y₂=. For a window, use the values 0 to 5 for $x$ and –10 to 10 for $y$. Press [GRAPH]. The graphs should intersect somewhere a little to right of $x=1$.

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, $x \approx 1.339$.

Try It #9

Solve $5log(x+2)=4−log(x)$ graphically. Round to the nearest thousandth.