Section 5-5:  Logarithmic Functions
 
 
Common Logarithm 

Demo: Log Funtion Applet
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!