10.2 Logarithms and Logarithmic Functions

Objectives:1. Evaluate logarithmic expressions.2. Solve logarithmic equations and

inequalities.

Logarithms

The inverse of is In y is called the logarithm of x,

usually written 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 denotedand it defined as the exponent y that makes the equation true.

xy b yx byx b

logby x

logb x

yb x

ExamplesWrite each equation in

exponential form.1.

2.

Write each equation in logarithmic form.

1.

2.

3log 9 223 9

10

1log 2

100

2 110

100

35 1255log 125 3

log expbase answer

1

327 3

27

1log 3

3

Logarithmic FunctionsThis function is the inverse of the exponential

function 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 numbers5. The graph contains the point (1,0). (The x-

intercept is 1)

xy b

Properties

The following is true for all logarithms:Ex:

logb xb x 5log ( 2)5 2x x

log xb b x 3

7log 7 3x x

Evaluating Log Expressions

Write in exponential form. Rewrite with like-bases, set exponents equal to each other.

Example:

Evaluate:

Evaluate:

3log 243 3 243x 5(3 243)53 3x 5x

29log 9 29 9x 2x

27log ( 1)7 x

2 1x

Logarithmic to Exponential Inequality

If b>1, x>0, and then x>If b>1, x>0, and then 0<x<

If x>

If

logb x y yblogb x y yb

3log 4x 43

5log 2x

20 5x

Property of equality for Log Functions

If b is a positive number other than 1, thenif and only if x=y.

Example: if x=10

log logb bx y

3 3log log 10x

Property of Inequality for Log Functions

If b>1 then if and only if x>y and if and only if x<y.

If then x<8

If then x>3

log logb bx ylog logb bx y

4 4log log 8x

6 6log log 3x

Examples

Solve.1.

2.

3.

8

4log

3n

4

38 n

16n

6 6log (2 3) log ( 2)x x 2 3 2

5

x x

x

210 10log ( 6) logx x 2 6x x 2 6 0x x

( 3)( 2) 0

3, 2

x x

x x

Homeworkp. 536

22-40 even48-60 even