Inverse and Log Functions
How to find inverse functions:
1. Take y  f (x)
2. Interchange x and y
3. Solve for y
Ex: Find the inverse function of f ( x)  x 3  2
Note: We express the inverse function as f 1 ( x) , this should not
be confused with
1
. “-1” is not an exponent.
f x 
The inverse graph is found by interchanging the x and y values.
The point a, b  becomes the point b, a  . We can also get this point
by reflecting the point over the line y  x .
Ex:
x
The exponential function f ( x)  a has an inverse function which
is called the logarithmic function with a base of a.
Exp: y  a
Note:
x  ay
Inverse/Log:
y  log a x
x
log a a x   x
→
x
x
since a  a
a loga x  x
→
y
since y  log a x and x  a
(or take log a of both sides)
Log Laws
1. log a xy   log a x  log a y
x
2. log a  y   log a x  log a y
 
m
3. log a x  m log a x
4. log a x 
log x
log a
Evaluate: log 2 80  log 2 5
Log graph
The graph of all log functions pass through the point (1,0)
Natural Logs
This is a log with a base of e.
log e x  ln x
Properties (follow the same log rules)
ln e x  x
eln x  x
ln e  1
Ex: Solve: e 3 x5  10
Graph the following graph: y  ln x  2  1