How do we solve these equations for x?a. b. c.

Solving Exponential Equations…

10010 x

4010 x

7510 x

Section 4.4 Logarithmic Functions

1. Definition2. Special Logarithms : Base 10 and Base e3. Change of Base formula4. Inverse Function5. Graphing the log function6. More log properties7. Solving Equations

The exponential form of is

The logarithmic form of is

I. Definition of Logarithmic Functions

A logarithmic function with base a is denoted:

where a > 0 and a ≠ 1 and is defined by

if and only if

y loga x

y loga xyax

What power of a gives you the value ?

The answer: xalog

x

xy alog xa y

xa y xy alog

Practice : Rewrite the expressionWrite the exponential expression into an equivalent logarithmic form.

x416 Ask: What power of 4 produces the number 16 ?

Answer: The logarithm of 16, base 4

Ask: What power of ______ produces the number ____?

Answer: The logarithm of ____, base _____.

x5125

Practice : Rewrite the expressionWrite the logarithmic expression into an equivalent exponential form.

x16log2

9log3

Find the exact value (without a calculator).

1log2

16log 2/1

3 raised to what power ___ equals 9 ?

2 raised to what power ___ equals 1 ?

½ raised to what power ___ equals 16 ?

2. Special Logarithms - Common Log

If a base is not indicated, it is understood to be 10.

yxxy 10 ifonly and if log

Example:

is equivalent to )40log(y 4010 y

2. Special Logarithms – Natural Log

natural logarithm function is expressed using the special symbol ln (logarithmus naturalis), instead of the log symbol

yexxy ifonly and if ln

5lnyExample:

is equivalent to 5ye

Practice: Solve exponential equations

and evaluate log expressions

Review: #58

p. 284, #59, 60, 62, 63,66 Solve exponential expressionp. 297 #33-37,39,41-43 Determine the value of the logarithmic expression.

10log 10/1

12log12

3. Change-of-Base Formula

log lnlog

log lnaM M

Ma a

Your calculator can compute only base 10 and base e.

Use this formula to get an approximation for a logarithm to base neither 10 nor e.

Example.

Find an approximation for )5(log2

4. Inverse Properties of Logarithmic and Exponential FunctionsThe Logarithmic and Exponential Functions

are inverses of each other.

Example of the relationship: Let xxfxg x2log)( and 2)(

Inverse Property of xxfbxg bx log)( and )(

xbxgfbxfg xb

xb log))(( andx ))(( log

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

f (x) ax

Exponential functions and log functions are inverse functions

of each other.

f 1(x) loga x

Domain:Range:Key Points:Asymptotes:

5. Graphing Logarithmic Functions

5. A) Domain of a logarithmic function

Determine the domain for these functions.

82log)( 23 xxf

1

1ln)(x

xf

12log)( 4 xxf

Practice: Worksheet 4.3-4.4

and p. 297 #46-56 (even),

and if M = N then

6) A Special Property of Logarithms

If then M = N

NM aa loglog

NM aa loglog

NOTE:We typically use base e

(natural log)when applying the log to

both sides of the equation.

7. Solving Logarithmic Equations.

Logarithmic Equations(always check answer against domain of problem)

1. Equal base on each side:

2. Constant on one side, logarithm on other:

)26(log)82(log 44 xx

15)82(log4 x

Use the property: then M = N NM aa loglog

Use the definition: xy alog xa y

CHECK IT! Logarithms are only defined for positive real numbers!Exclude solutions that produce logarithm of a number

Exponential Equations.

1. Equal base on each side:

2. Constant on one side, exponential on other. Two ways to solve. a) Method 1 :

b) Method 2:

3. Quadratic in form:

8. Solving Exponential Equations.

2

22 43 xx

43 12

x

43 xe

0342 xx ee

Use the property:

If au av, then uv

Use the definition: xy alogxa y

Take ln (natural log) of each side.We will look at this method after Section 4.5

9. ApplicationThe formula

models the population of Florida, A, in millions, x years after 2000. Suppose the population is 16.3 million in 2001.

a) Determine the population of Florida in the year 2010.

b) When will the population reach 25.2 million ?

kxeA 9.15