Logarithmic Functions Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights...

Logarithmic Functions

Digital Lesson

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

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 = 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


log10 –4 LOG –4 ENTER ERROR

no power of 10 gives a negative number

The base 10 logarithm function f (x) = log10 x is called the

common logarithm function.

The LOG key on a calculator is used to obtain common logarithms.

Examples: Calculate the values using a calculator.

log10 100

log10 5

Function Value Keystrokes Display

LOG 100 ENTER 2

LOG 5 ENTER 0.69897005

2log10( ) – 0.3979400LOG ( 2 5 ) ENTER


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 a

x = x and alogax = x inverse property


x

y

Graph f (x) = log2 x

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


Example: Graph the common logarithm function f(x) = log10 x.

by calculator

1

10

10.6020.3010–1–2f(x) = log10 x

10421x 1

100

y

x5

–5

f(x) = log10 x

x = 0 vertical

asymptote

(0, 1) x-intercept


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

3. x-intercept (1, 0)

5. increasing6. continuous7. 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

y y = x

y = log2 x

y = a x

domain

range

y-axisvertical

asymptote

x-intercept(1, 0)


The function defined by f(x) = loge x = ln x

is called the natural logarithm function.

Use a calculator to evaluate: ln 3, ln –2, ln 100

ln 3ln –2ln 100

Function Value Keystrokes Display

LN 3 ENTER 1.0986122ERRORLN –2 ENTER

LN 100 ENTER 4.6051701

y = ln x

(x 0, e 2.718281)

y

x5

–5

y = ln x is equivalent to e y = x


Properties of Natural Logarithms 1. ln 1 = 0 since e0 = 1. 2. ln e = 1 since e1 = e. 3. ln ex = x and eln x = x inverse property 4. If ln x = ln y, then x = y. one-to-one property

Examples: Simplify each expression.

2

1ln

e 2ln 2 e inverse property

20lne 20 inverse property

eln3 3)1(3 property 2

00 1ln property 1


Example: The formula (t in years) is used to estimate

the age of organic material. The ratio of carbon 14 to carbon 12

in a piece of charcoal found at an archaeological dig is .

How old is it?

82231210

1 t

eR

To the nearest thousand years the charcoal is 57,000 years old.

original equation158223

12 10

1

10

1

t

e

multiply both sides by 1012

1000

18223 t

e

take the natural log of both sides1000

1lnln 8223

t

e

inverse property1000

1ln

8223

t

56796907.6 82231000

1ln 8223

t

1510

1R

Logarithmic Functions Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights...

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