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6.4 Logarithmic Functions

In this section, we will study the following topics:

Evaluating logarithmic functions with base a

Graphing logarithmic functions with base a

Evaluating and graphing the natural logarithmic function

Solving logarithmic and exponential equations

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Logarithmic Functions

Now that you have studied the exponential function, it is time to take a look at its INVERSE: THE LOGARITHMIC FUNCTION.

In the exponential function, the independent variable (x) was the exponent. So we substituted values into the exponent and evaluated it for a given base.

Exponential Function: f(x) = 2x, f(3) = 23 = 8.

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Logarithmic Functions

For the inverse function (LOGARITHMIC FUNCTION), the base is given and the answer is given, so to evaluate a logarithmic function is to find the exponent.

That is why I think of the logarithmic function as the “Guess That Exponent” function.

Warm Up: Give the value of ? in each of the following equations.? ? ?11) 3 81 2) 5 3) 16 4

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Exponential and logarithmic functions of the same base are inverses.

Exponential and logarithmic functions of the same base are inverses.

Exponential and logarithmic functions of the same base are inverses.

Exponential and logarithmic functions of the same base are inverses.

Exponential and logarithmic functions of the same base are inverses.

Exponential and logarithmic functions of the same base are inverses.

Exponential and logarithmic functions of the same base are inverses.

Exponential and logarithmic functions of the same base are inverses.

Exponential and logarithmic functions of the same base are inverses.

Subliminal Message

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Logarithmic Functions (continued)

Evaluate log28

To evaluate log28 means to find the exponent such that 2 raised to

that power gives you 8.

?

2log 8 ?

2 8

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Logarithmic Functions (continued)

The following definition demonstrates this connection between the exponential and the logarithmic function.

Definition of a Logarithmic FunctionFor x > 0, a > 0, and a ≠ 1,

y = logax if and only if x = ay

We read logax as “log base a of x”.

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Converting Between Exponential and Logarithmic Forms

I. Write the logarithmic equation in exponential form.

a)

b)

II. Write the exponential equation in logarithmic form.

a)

b)

3log 81 4

71log 249

329 27

2 1864

y = logax if and only if x = ay

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Evaluating Logarithms w/o a Calculator

To evaluate logarithmic expressions by hand, we can use the related

exponential expression.

Example:

Evaluate the following logarithms:

10 5110,000 b)) log log

25a

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Evaluating Logarithms w/o a Calculator (cont.)

336) 6 d) log 1logc

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Evaluating Logarithms w/o a Calculator

Okay, try these.

e) f)

g) h)

5log 5 4log 0

81log2 10log 0.0001

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The Common Logarithm

The common logarithm has a base of 10. If the base

of a logarithm is not indicated, then it is assumed

that the base is 10.

10For example, log 0.01 is equivalent to log 0.01

log if and only if 10 yy x x

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Graphs of Logarithmic Functions

Since the logarithmic function is the _______________ of the

exponential function (with the same base), we can use what

we know about inverse functions to graph it.

Example:

Graph f(x) = 2x and g(x) = log2x in the same coordinate plane.

To do this, we will make a table of values for f(x)=2x and then

switch the x and y coordinates to make a table of values for g(x).

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Graphs of Logarithmic Functions (continued)

f(x) = 2x g(x) = log2x

x f(x)

-4

-2

0

2

4

x g(x)

f(x) = 2x

g(x)= log2x

y =x

Inverse functions

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Comparing the Graphs of Exponential and Log Functions

Notice that the domain and range of the inverse functions are switched.

The exponential function has

domain (-, )

range (0, )

HORIZONTAL asymptote y = 0

The logarithmic function has

domain (0, )

Range (-, )

VERTICAL asymptote x = 0

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Transformations of Graphs of Logarithmic Functions

The same transformations we studied earlier also apply to logarithmic functions. Look at the following shifts and reflections of the graph of f(x) = log2x.

The new vertical asymptote is x = -2

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Transformations of Graphs of Logarithmic Functions

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The Natural Logarithmic Function

In section 6.3, we saw the natural exponential function with base

e. Its inverse is the natural logarithmic function with base e.

Instead of writing the natural log as logex, we use the

notation , which is read as “the natural log of x” and is

understood to have base e.

ln x

ln if and only if yy x x e

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Natural Log Key

To evaluate the natural log using the TI-83/84, use the button.

Notice, the 2nd function of this key is ________.

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Graph of the Natural Exponential and Natural Logarithmic Function

f(x) = ex and g(x) = ln x are inverse functions and, as such, their graphs are reflections of one another in the line y = x.

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Evaluate without using a calculator.

a) b)

c) d)

e) f)

3ln e

5ln eln ( 2)

4ln e

1lne

ln1

Evaluating the Natural Log

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Solving Logarithmic Equations

Strategy for solving logarithmic equations:

Change the equation from a log equation into an exponential equation, using one of the following forms: logax = y x = ay

logx = y x = 10y

lnx = y x = ey

Keep in mind that the domain of the log function is x>0. Reject any extraneous solutions!!

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Examples of solving log equations

51) log 3x 32) log 3 2 2x

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More examples of solving log equations

63) log 36 5 3x 254) log 4 2x x

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End of Section 6.4