Logarithmic Functions & Graphs, Lesson 3.2, page 388

Objective: To graph logarithmic functions, to convert between exponential and logarithmic

equations, and find common and natural logarithms using a

calculator.

DEFINITION Logarithmic function – inverse of

exponential function If y = bx, then the inverse is x = by

So y is the power which we raise b to in order to get x.

Since we can’t solve this for y, we change it to logarithmic form which is

y = logbx

Think of logs like this…logbN = P and bp = N

Key: b = base, N = number, P = power

Restrictions: b > 0 and b cannot equal 1

*N > 0 because the log of zero or a negative number is undefined.

Changing Exponential Log

Log form => logb N = P Ex) log28 = 3

Think: A logarithm equals an exponent!

Exponential form => bP = N Ex) 23 = 8

Examples of ConversionLog Form: logbN = P Exponential Form: bP

= N

Log264 = 6 Log101000 = 3 Log416 = 2

25 = 32 104 = 10000 44 = 256

Rewrite the following exponential expression as a logarithmic one.

7

3

3

3

) log ( 2) 3) log ( 2) 7) log (7) 2) log ( 2) 7

a xb xc xd x

73 )2( x

See Example 1, page 389.Check Point 1.

Write each equation in its equivalent exponential form:

A) 3 = log7x B) 2 = logb25

C) log426 = y

See Example 2, page 389.Check Point 2.

Write each equation in its equivalent logarithmic form:

A) 25 = x B) b3 = 27

C) e y = 33

See Example 3, page 389.Check Point 3.

Evaluate: A) log10 100 B) log3 3

C) log36 6

See page 390.BASIC LOG PROPERTIES logb b = 1 logb 1 = 0INVERSE PROPERTIES OF LOGS logb bx = x blogbx = x

ExamplesCheck Point 4. A) log99 b) log8 1

Check Point 5: A) log7 78 b) 3log317

Graphs Since exponential and logarithmic

functions are inverses of each other, their graphs are also inverses.

Logarithmic function and

exponential function are inverses of each other.

The domain of the exponential function is all reals, so that’s the domain of the logarithmic function.

The range of the exponential function is x>0, so the range of the logarithmic function is y>0.

log

logb

x

x

b

b

x

x

b

See Example 6, page 391.Check Point 6: Graph f(x) = 3x and g(x) = log3 x in

the same rectangular coordinate system.

Graph f(x) = 3x.

(3,1/27)1/273

(2, 1/9)1/92

(1, 1/3)1/31

(3, 27)273

9

3

1

y = f(x) = 3x

(2, 9)2

(1, 3)1

(0, 1)0

(x, y)x

Now let’s add f(x) = log3x.(Simply find the inverse of each point from

f(x)= 3x.)

f(x)= 3x

(0, 1)

(1, 3)

(2, 9)

(3, 27)

(1, 1/3)

(2, 1/9)

(3,1/27)

xxf 3log)(

See Characteristics of Graphs of Logs on page 392. See Table 3.4 on Transformations.

Graphing Summary

Logarithmic functions are inverses of exponential functions. Easier if rewrite as an exponential before graphing.

1. Choose values for y.2. Compute values for x.3. Plot the points and connect them with a smooth curve.

* Note that the curve does not touch or cross the y-axis.

Comparing Exponential and Logarithmic Functions

Domain Restrictions for Logarithmic Functions

Since a positive number raised to an exponent (pos. or neg.) always results in a positive value, you can ONLY take the logarithm of a POSITIVE NUMBER.

Remember, the question is: What POWER can I raise the base to, to get this value?

DOMAIN RESTRICTION: 0,log xxy b

See Example 7, page 393. Check Point 7: Find the domain of

f(x)=log4 (x-5).

Common Logarithms -- Intro

If no value is stated for the base, it is assumed to be base 10.

log(1000) means, “What power do I raise 10 to, to get 1000?” The answer is 3.

log(1/10) means, “What power do I raise 10 to, to get 1/10?” The answer is -1.

COMMON LOGARITHMS A common logarithm is a log that

uses 10 as its base. Log10 y is written simply as log y.

Examples of common logs areLog 100, log 50, log 26.2, log (1/4)

Log button on your calculator is the common log *

Find each of the following common logarithms on a calculator.

Round to four decimal places.a) log 723,456b) log 0.0000245c) log (4)

Find each of the following common logarithms on a calculator.

Does not existERR: non real anslog (4)4.61084.610833916log 0.00002455.85945.859412123log 723,456

RoundedReadoutFunction Value

Natural Logarithms -- Intro ln(x) represents the natural log of x, which

has a base=e What is e? If you plug large values into

you get closer and closer to e. logarithmic functions that involve base e are

found throughout nature Calculators have a button “ln” which

represents the natural log.

x

x

11

)ln(log xxe

Natural Logarithms

Logarithms, base e, are called natural logarithms.

The abbreviation “ln” is generally used for natural logarithms.

Thus, ln x means loge x.

* ln button on your calculator

is the natural log *

Find each of the following natural logarithms on a calculator.

Round to four decimal places.a) ln 723,456b) ln 0.0000245c) ln (4)

Does not existERR: nonreal answerln (4)

10.616810.61683744ln 0.0000245

13.491813.49179501ln 723,456

RoundedReadoutFunction Value