Transformations of Graphs of Logarithmic Functions

Translations of Logarithmic Functions

All translations of the parent logarithmic function,  $y=log_b(x)$,  have the form

$f(x)=alog_b(x+c)+d$

where the parent function, $y=log_b(x),b > 1$, is

• shifted vertically up $d$ units.
• shifted horizontally to the left $c$ units.
• stretched vertically by a factor of $|a|$ if $|a| > 0$.
• compressed vertically by a factor of $|a|$ if $0 < |a| < 1$.
• reflected about the $x$-axis when $a < 0$.

For $f(x)=log(−x)$, the graph of the parent function is reflected about the $y$-axis.

Example 10

Finding the Vertical Asymptote of a Logarithm Graph

What is the vertical asymptote of $f(x)=−2log_3(x+4)+5$?

Solution

The vertical asymptote is at $x=−4$.

Analysis

The coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve 4 units to the left shifts the vertical asymptote to $x=−4$.

Try It #10

What is the vertical asymptote of $f(x)=3+ \ln(x−1)$?

Example 11

Finding the Equation from a Graph

Find a possible equation for the common logarithmic function graphed in Figure 15.

Figure 15

Solution

This graph has a vertical asymptote at $x=–2$ and has been vertically reflected. We do not know yet the vertical shift or the vertical stretch. We know so far that the equation will have form:

$f(x)=−alog(x+2)+k$

It appears the graph passes through the points $(–1,1)$ and $(2,–1)$. Substituting $(–1,1)$,

$\begin{array}{ll} 1=-a \log (-1+2)+k & \text { Substitute }(-1,1) \\ 1=-a \log (1)+k & \text { Arithmetic. } \\ 1=k & \log (1)=0 \end{array}$

Next, substituting in $(2,–1)$,

$\begin{aligned} -1 &=-a \log (2+2)+1 & & \text { Plug in }(2,-1) \\ -2 &=-a \log (4) & & \text { Arithmetic. } \\ a &=\frac{2}{\log (4)} & & \text { Solve for } a \end{aligned}$

This gives us the equation $f(x)=–\frac{2}{log(4)}log(x+2)+1$.

Analysis

We can verify this answer by comparing the function values in Table 5 with the points on the graph in Figure 15.

$x$	$-1$	$0$	$1$	$2$	$3$
$f(x)$	$1$	$0$	$−0.58496$	$-1$	$−1.3219$
$x$	$4$	$5$	$6$	$7$	$8$
$f(x)$	$−1.5850$	$−1.8074$	$-2$	$−2.1699$	$−2.3219$

Table 5

Try It #11

Give the equation of the natural logarithm graphed in Figure 16.

Figure 16

Q&A

Is it possible to tell the domain and range and describe the end behavior of a function just by looking at the graph?

Yes, if we know the function is a general logarithmic function. For example, look at the graph in Figure 16. The graph approaches $x=−3$ (or thereabouts) more and more closely, so $x=−3$ is, or is very close to, the vertical asymptote. It approaches from the right, so the domain is all points to the right, $\{ x|x > −3 \}$. The range, as with all general logarithmic functions, is all real numbers. And we can see the end behavior because the graph goes down as it goes left and up as it goes right. The end behavior is that as $x→−3+,f(x)→− \infty$ and as $x→ \infty,f(x)→ \infty$.