Exponential and Logarithmic Functions
5.5 Graphs of Logarithmic Functions
Core Log Graph Features
Log graphs are inverse images of exponential graphs.
Fact: Features of $f(x)=\log_b(x)$ with $b>1$
- Domain: $(0,\infty)$
- Range: $(-\infty,\infty)$
- Vertical asymptote: $x=0$
- x-intercept: $(1,0)$
Example: Identify graph features
For
$$f(x)=\log_4(x),$$
state domain, range, and asymptote.
Show Solution
- Domain: $(0,\infty)$
- Range: $(-\infty,\infty)$
- Vertical asymptote: $x=0$
MyOpenMath: Practice: logarithm graph features (placeholder)
Transforming Log Graphs
Transformations apply the same way as other functions, with domain checks first.
Example: Analyze a transformed log
Describe
$$g(x)=\log_2(x-3)+1.$$
Show Solution
Start with $\log_2(x)$.
- $x-3$ shifts graph right 3.
- $+1$ shifts graph up 1.
Vertical asymptote shifts from $x=0$ to
$$x=3.$$
Domain is
$$(3,\infty).$$
