1. THE LOGARITHMIC FUNCTION
5.3
THE LOGARITHMIC FUNCTION

2. The Graph, Domain, and Range of the Common Logarithm
The domain of log x is all positive numbers. Its range is all real numbers. The log function grows very rapidly for 0 < x < 1 and very slowly for x > 1. It has a vertical asymptote at x = 0 and never touches the y-axis.

3. Graphs of the Inverse Functions y = log x and y = 10x
Log function x
y = log x
0.01 -2
0.1 -1 1 10
100 2
1000 3
y = x
(1,10 )
y = 10x
Exponential function x
y = 10x -2
0.01 -1
0.1 1 10 2
100 3
1000
(0,1 )
(10,1 )
y = log x
(-1,0.1 )
(1,0 )
(0.1,-1 )

4. Graph of the Natural Logarithm
y = ln x
Like the common log, the natural log is only defined for x > 0 and has a vertical asymptote at x = 0. The graph is slowly increasing and concave down. It also passes through (1, 0).

5. Graph of Natural Logarithm and Its Inverse
y = ex
The functions y = ln x and y = ex are inverses of one another. Notice how they are mirror images of one another through the line y = x.
y = x
y = ln x