Logarithmic Functions

Objectives

Recognize and evaluate logarithmic functions with base a.

Graph logarithmic functions with base a.

Recognize, evaluate, and graph logarithmic functions with base e.

Definition: Logarithmic Function

For x 0 and 0 a 1, y = loga x if and only if x = a y.

The function given by f (x) = loga x is called the

logarithmic function with base a.

Every logarithmic equation has an equivalent exponential form: y = loga x is equivalent to x = a y

A logarithmic function is the inverse function of an exponential function.

Exponential function: y = ax

Logarithmic function: y = logax is equivalent to x = ay

A logarithm is an exponent!

y = logax if and only if x = a y

The logarithmic function to the base a, where a > 0 and a 1 is defined:

2416

exponential formlogarithmic form

Convert to log form: 216log4 Convert to exponential form:

38

1log2

8

12 3

When you convert an exponential to log form, notice that the exponent in the exponential becomes what the log is equal to.

LOGS = EXPONENTS

With this in mind, we can answer questions about the log:

16log2

This is asking for an exponent. What exponent do you put on the base of 2 to get 16? (2 to the what is 16?)4

9

1log3

What exponent do you put on the base of 3 to get 1/9? (hint: think negative)2

1log4

What exponent do you put on the base of 4 to get 1?

0

3log3

When working with logs, re-write any radicals as rational exponents.

2

1

3 3logWhat exponent do you put on the base of 3 to get 3 to the 1/2? (hint: think rational)2

1

Examples: Write Equivalent Equations

y = log2( )2

1= 2

y

2

1

Examples: Write the equivalent exponential equation and solve for y.

1 = 5 yy = log51

16 = 4y y = log416

16 = 2yy = log216

SolutionEquivalent Exponential

Equation

Logarithmic Equation

16 = 24 y = 4

2

1= 2-1 y = –1

16 = 42 y = 2

1 = 50 y = 0

Your Turn:• Write each equation in exponential form

log 3 81 = 434=81

log 7 1/49 = -27-2=1/49

• Write each equation in logarithmic form103 = 1000

Log101000=34-2 = 1/16

Log41/16=-2

Your Turn:

Find y in each equation.• log 2 8 = y

y=3

• log 5 1 = y

25

y=-2

Properties of Logarithms

Examples: Solve for x: log6 6 = x

log6 6 = 1 property 2 x = 1

Simplify: log3 35

log3 35 = 5 property 3

Simplify: 7log7

9

7log7

9 = 9 property 3

Properties of Logarithms

1. loga 1 = 0 since a0 = 1.

2. loga a = 1 since a1 = a.

4. If loga x = loga y, then x = y. one-to-one property

3. loga ax = x and alogax = x inverse property

Your Turn:

Solve

1. 0

2. 1

3. 30

5log 1

11log 11

8log 308

Base 10 logarithms

• Called common logarithms• When base a is not indicated, it is

understood that a = 10• log 1/100 = log 10 = • log 1/10 = log 100 =• log 1 = log 1000 =• The LOG key on your calculator.

-2

-1

0

1

2

3

In the last section we learned about the graphs of exponentials.

Logs and exponentials are inverse functions of each other so let’s see what we can tell about the graphs of logs based on what we learned about the graphs of exponentials.

Recall that for functions and their inverses, x’s and y’s trade places. So anything that was true about x’s or the domain of a function, will be true about y’s or the range of the inverse function and vice versa.

Let’s look at the characteristics of the graphs of exponentials then and see what this tells us about the graphs of their inverse functions which are logarithms.

Characteristics about the Graph of an Exponential Function a > 1

xaxf 1. Domain is all real numbers

2. Range is positive real numbers

3. There are no x intercepts because there is no x value that you can put in the function to make it = 0

4. The y intercept is always (0,1) because a 0 = 1

5. The graph is always increasing

6. The x-axis (where y = 0) is a horizontal asymptote for x -

Characteristics about the Graph of a Log Function a > 1

xxf alog1. Range is all real numbers

2. Domain is positive real numbers

3. There are no y intercepts

4. The x intercept is always (1,0) (x’s and y’s trade places)

5. The graph is always increasing

6. The y-axis (where x = 0) is a vertical asymptote

Exponential Graph Logarithmic Graph

Graphs of inverse functions are reflected about the line y = x

basea>1

Graph f(x) = log2 x

x

y

Since the logarithm function is the inverse of the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x.

83

42

21

10

–1

–2

2xx

4

1

2

1

y = log2 x

y = xy = 2x

(1, 0)

x-intercept

horizontal asymptote y = 0

vertical asymptote x = 0

Graphs of Logarithmic Functions

The graphs of logarithmic functions are similar for different values of a. f(x) = loga x (a 1)

3. x-intercept (1, 0)

5. increasing

6. continuous

7. one-to-one

8. reflection of y = a x in y = x

1. domain ),0( 2. range ),(

4. vertical asymptote

)(0 as 0 xfxx

Graph of f (x) = loga x (a 1)

x

yy = x

y = log2 x

y = a x

domain

range

y-axisverticalasymptote

x-intercept(1, 0)

Graphs of Logarithmic Functions

• Typical shape for graphs where a > 1 (includes base e and base 10 graphs).

• Typical shape for graphs where 0 < a < 1.

The Logarithmic Function: f (x) = loga x, a > 1

The Logarithmic Function: f (x) = loga x, 0 < a < 1

Determining Domains of Logarithmic Functions

Example Find the domain of each function.

Solution

(a) Argument of the logarithm must be positive.

x – 1 > 0, or x > 1. The domain is (1,).(b) Use the sign graph to solve x2 – 4 > 0.

)4ln()( (b) )1(log)( (a) 22 xxfxxf

The domain is (–,–2) (2, ).

Your Turn:

Find the domain.

(a)

Solutiona) Domain: (-3,) or x >-3b) Domain: (-3, 3) or -3<x <3

22 3log ( 3) (b) log 9y x y x

• Any log to the base e is known as a

natural logarithm.• In French this is a

logarithme naturel• Which is where ln comes from.• When you see ln (instead of log)

– then it’s a natural log• y = ln x is the inverse of y = ex

• The LN key on your calculator.

Natural Logarithms

Natural Logarithms

Natural Logarithms

Natural Logarithms

Natural Logarithms