Section 5-5: Logarithmic
Functions
Common
Logarithm 
log x = a if and only
if 10a = x
The important thing to
remember is the log represents the exponent. In the case of common
logs, the base is always base 10. Study the following examples.
1) log 100 = 2
because 102 = 100.
2) log 1000 = 3
because 103 = 1000.
3) log 1 = 0 because
100 = 1.
4) log .1 = -1
because 10-1 = .1
5) log .01 = -2
because 10-2 = .01
The log function is the
inverse function of the exponential function and as such their graphs are
reflections about the y = x line. Here is the graph of the common
log and the inverse.
Some important facts
you need to understand from the log graph. The domain of the log
is x > 0. The range is all real numbers. The zero is at x =
1. You can only find the log of positive numbers. Logs of numbers
less than one are negative and logs of numbers greater than one are positive.
We can find the log of
other bases by using the following formula similar to the common log definition.
logb
x = n if and only if x = bn.
Here are some examples:
1) log2
8 = 3 because 23 =
8
2) log3
81 = 4 because 34
= 81.
3) log4
1/16 = -2 because 4-2
= 1/16
4) log8
1 = 0 because 80 =
1
One of the most important
log function is called the natural log which has the number e as the base.
When e is used as a base we use the following notation:
ln x = a if and only
if ea = x
Most natural logs need
to be calculated on your calculator. The graph of the natural log
is shown below:
Solving Simple Log Equations
1) Log x = 3
Solution: To solve an equation of this
type, rewrite the equation in exponential form. x = 103
= 1000
2) Log |x| = 2
Solution: To solve an equation of this
type, again rewrite the equation in exponential form and solve for x.
|x| = 102
= 100
x = 100 or -100
3) Log (x2
+ 19) = 2
Solution: Again, rewrite as an exponential
equation and solve for x.
x2 + 19 =
102
x2 + 19 =
100
x2 = 81
x = 9 or -9
4) Log x = .3
Again, rewrite exponentially.
x = 10.3
Use your calculator and round to hundredths.
x = 2.00
5) Ln x = -1.2
Solution: Same as above.
x = e-1.2
x = .30
On to the Law of Logs: 
Please let me back up
and regroup! 