1. Logarithmic Functions
Section 5.5:
Logarithmic Functions

2. log x = a if and only if 10a = x
common logarithm – the common logarithm of any positive real number x is defined to be the exponent you get when you write x as a power of 10.
log x = a if and only if 10a = x

3. 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

4. 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.

5. 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.

6. logb x = n if and only if x = bn
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

7. 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 on the next slide.

9. Example 1:
Evaluate.
a) log82
Let the log82 = x
8x = 2
(23)x = 21
23x = 21
3x = 1
x = 1/3

10. ln 1 e3
ln 1 = x
ex = 1
ex = e-3
x = -3

11. log
10,000
x = -4
log51 = x
x = 0

12. Example 2:
Solve.
a) log x = 4
x = 104
x = 10,000

13. ln x = ½
e1/2 = x
x ≈ 1.65
c) log x = -1.2
= x
x ≈ 0.063

14. HOMEWORK
p. 194 – 195; 12 – 20 even