*Section 5-5:
Logarithmic
Functions*
* *
*Common
Logarithm *

Demo:
Log Funtion Applet
*log x = a if and
only
if 10*^{a} = 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 10*^{2} = 100.
*2) log 1000 =
3
because 10*^{3} = 1000.
*3) log 1 = 0
because
10*^{0} = 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.*
*log*_{b}
x = n if and only if x = b^{n}.
*Here are some
examples:*
*1) log*_{2}
8 = 3 because 2^{3}
=
8
*2) log*_{3}
81 = 4 because 3^{4}
= 81.
*3) log*_{4}
1/16 = -2 because 4^{-2}
= 1/16
*4) log*_{8}
1 = 0 because 8^{0}
=
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 e*^{a} = 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 = 10*^{3}
= 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| = 10*^{2}
= 100
*x = 100 or -100*
* *

*3) Log (x*^{2}
+ 19) = 2

*
Solution: Again, rewrite as an
exponential
equation and solve for x.*
*x*^{2} + 19
=
10^{2}
*x*^{2} + 19
=
100
*x*^{2} = 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! *
* *