Download 10.2 Logarithms and Logarithmic Functions

10.2 Logarithms and Logarithmic
Functions
Objectives:
1. Evaluate logarithmic expressions.
2. Solve logarithmic equations and
inequalities.
Logarithms
The inverse of y  b is x  b
y
In x  b y is called the logarithm of x, usually
written y  logb x read “y equals log base
b of x. Logs are a shortcut to solving for x or y.
Let b and x be positive numbers, b≠1. The
logarithm of x with base b is denoted log b x
and it defined as the exponent y that makes
y
the equation b  x true.
x
y
Examples
Write each equation in
exponential form.
1. log3 9  2
Write each equation in
logarithmic form.
1. 53  125
log5 125  3
3 9
2
2.
1
log10
 2
100
1
2
10 
100
logbase answer  exp
2.
1
3
27  3
1
log 27 3 
3
Logarithmic Functions
This function is the inverse of the exponential
x
function y  b It has the following
characteristics:
1. The function is continuous and one-to-one.
2. The domain is the set of all positive real
numbers.
3. The y-axis is an asmyptote of the graph.
4. The range is the set of all real numbers
5. The graph contains the point (1,0). (The xintercept is 1)
Properties
The following is true for all logarithms:
Ex: 5log5 ( x  2) 
logb x
b
x
log b b  x
x
log 7 7
x 3
x2
 x 3
Evaluating Log Expressions
Write in exponential form. Rewrite with likebases, set exponents equal to each other.
5
x
Example: log3 243 3  243 (3  243)
3  23
x
Evaluate: log 9 9
Evaluate:
7
x5
5
9 9
x
log7 ( x 2 1)
x 1
2
2
x2
Logarithmic to Exponential Inequality
If b>1, x>0, and logb x  y then x>b
y
If b>1, x>0, and logb x  y then 0<x<b
y
If log3 x  4
x> 34
If log5 x  2
0 x5
2
Property of equality for Log Functions
If b is a positive number other than 1, then
logb x  logb y if and only if x=y.
Example: if log3 x  log3 10 x=10
Property of Inequality for Log
Functions
If b>1 then logb x  logb y if and only if x>y
and logb x  logb y if and only if x<y.
If log 4 x  log 4 8 then x<8
If log 6 x  log 6 3 then x>3
Examples
Solve.
1.
4
log 8 n 
3
4
3
8 n
n  16
2. log 6 (2 x  3)  log 6 ( x  2)
2x  3  x  2
x5
3. log10 ( x  6)  log10 x
2
x 6  x
2
x  x6  0
2
( x  3)( x  2)  0
x  3, x  2
Homework
p. 536
22-40 even
48-60 even