Transcript Algebra 2 Lesson 8

8.3 – Logarithmic
Functions and Inverses
What is a logarithm?
A logarithm is the power to which a number must be
raised in order to get some other number
If y  b , then log b y  x
x
For example, the base ten logarithm of 100 is 2, because
ten raised to the power of two is 100:
100 = 102 because
log 10100 = 2
Convert
If y  b x , then
If log b y  x then
If 7  3x , then
If log 4 64  3 then
Ex: Write the following equations in logarithmic form
Remember: If y = bx then logby = x
If 25 =
52
If 729 =
36
If 1 =
100
then
then
then
Log525=2
Log3729=6
Log101=0
If
3
1 1
  
2 8
then
1
log 1 ( )  3
2 8
Let’s try some:
Converting between the two forms.
Expo Form
3x  7
Log Form
log 3 7  x
10 2  100
log10 100  2
4 3  64
log 4 64  3
52 
2
5
1
25

1
32
log5
log 2
1
25
 2
1
 5
32
32  9
log3 9  2
10 3  1000
log 10 1000  3
Common Logs
 A common log is a logarithm that uses base 10.
You can write the common logarithm log10y as
log y (they are the logs you use on your
calculator)
 Scientists use common logarithms to measure
acidity, which increases as the concentration of
hydrogen ions in a substance. The pH of a
substance equals  log H  , where H  is the concentrat ion
of hydrogen ions.
Evaluating Logarithms
Ex: Evaluate log816
Log816=x
16 = 8x
24 = (23)x
24 = 23x
4 = 3x
x=4/3
Write an equation in log form
Convert to exponential form
Rewrite using the same base. In
this case, base of 2
Power of exponents
Set the exponents equal to each other
Solve for x
Therefore, Log816=4/3
Evaluating Logarithms
Ex: Evaluate log 64
log 64
1
32
1
x
Write an equation in log form
32
1
 64 x
32
Convert to exponential form
1
6x
5
6 x Rewrite using the same base. In

2

2

2
25
this case, base of 2. Use negative expos!
-5 = 6x
x=-5/6
Set the exponents equal to each other
Solve for x
1
5

Therefore,log 64
32
6
Let’s try some
Evaluate the following:
log 9 27
log 10 100