1 Copyright © Cengage Learning. All rights reserved. 5. Inverse, Exponential and Logarithmic...

1Copyright © Cengage Learning. All rights reserved.

5. Inverse, Exponential and

Logarithmic Functions

5.4 Logarithmic Function

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The exponential function given by for f (x) = ax for 0 < a < 1or a > 1 is one-to-one. Hence, f has an inverse function f

–1.

This inverse of the exponential function with base a is called the logarithmic function with base a and is denoted by loga.

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Its values are written loga (x) or loga x, read “the logarithm of x with base a.” Since, by the definition of an inverse function f

–1,

y = f –1(x) if and only if x = f (y),

the definition of loga may be expressed as follows.

Note that the two equations in the definition are equivalent. We call the first equation the logarithmic form and the second the exponential form.

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You should strive to become an expert in changing each form into the other.

The following diagram may help you achieve this goal.

Observe that when forms are changed, the bases of the logarithmic and exponential forms are the same.

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The number y (that is, loga x) corresponds to the exponent in the exponential form.

In words, loga x is the exponent to which the base a must be raised to obtain x.

This is what people are referring to when they say “Logarithms are exponents.”

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The following illustration contains examples of equivalent forms.

Illustration: Equivalent Forms

Logarithmic form Exponential form

• log5 u = 2 52 = u

• logb 8 = 3 b3 = 8

• r = logp q pr = q

• w = log4 (2t + 3) 4w = 2t + 3

• log3 x = 5 + 2z 35 + 2z = x

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The next example contains an application that involves changing from an exponential form to a logarithmic form.

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Example 1 – Changing exponential form to logarithmic form

The number N of bacteria in a certain culture after t hours is given by N = (1000)2t. Express t as a logarithmic function of N with base 2.

Solution:

N = (1000)2t given

isolate the exponential expression

change to logarithmic form

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The following general properties follow from the interpretation of loga x as an exponent.

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The reason for property 4 follows directly from the definition of loga, since

if y = loga x, then x = ay, or

The logarithmic function with base a is the inverse of the exponential function with base a, so the graph of y = loga x can be obtained by reflecting the graph of y = ax through the line y = x.

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This procedure is illustrated in Figure 1 for the case a > 1.

Figure 1

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Note that the x-intercept of the graph is 1, the domain is the set of positive real numbers, the range is , and the y-axis is a vertical asymptote.

Logarithms with base 0 < a < 1 are seldom used, so we will not emphasize their graphs.

We see from Figure 1 that if a > 1, then loga x is increasing on (0, ) and hence is one-to-one.

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When using this theorem as a reason for a step in the solution to an example, we will state that logarithmic functions are one-to-one.

In the following example we solve a simple logarithmic equation—that is, an equation involving a logarithm of an expression that contains a variable.

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Extraneous solutions may be introduced when logarithmic equations are solved.

Hence, we must check solutions of logarithmic equations to make sure that we are taking logarithms of only positive real numbers; otherwise, a logarithmic function is not defined.

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Example 3 – Solving a logarithmic equation

Solve the equation log6 (4x – 5) = log6 (2x + 1).

Solution:

log6 (4x – 5) = log6 (2x + 1)

(4x – 5) = (2x + 1)

2x = 6

x = 3

given

logarithmic functions are one-to-one

subtract 2x; add 5

divide by 2

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Example 3 – Solution

Check x = 3 LS: log6 (4 3 – 5) = log6 7

RS: log6 (2 3 + 1) = log6 7

Since log6 7 = log6 7 is a true statement, x = 3 is a solution.

cont’d

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When we check the solution x = 3 in Example 3, it is not required that the solution be positive.

But it is required that the two expressions, 4x – 5 and 2x + 1, be positive after we substitute 3 for x.

If we extend our idea of argument from variables to expressions, then when checking solutions, we can simply remember that arguments must be positive.

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Example 5 – Sketching the graph of a logarithmic function

Sketch the graph of f if f (x) = log3 x.

Solution:

We will describe three methods for sketching the graph.

Method 1:

Since the functions given by log3 xand 3x are inverses of each other,we proceed as we did for y = loga x inFigure 1; that is, we first sketch thegraph of y = 3x and then reflect itthrough the line y = x. Figure 1

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This gives us the sketch in Figure 2. Note that the points (–1, 3–1), (0, 1), (1, 3), and (2, 9) on the graph of y = 3x

reflect into the points (3–1, –1), (1, 0), (3, 1), and (9, 2) on the graph of y = log3 x.

Figure 2

cont’d

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Method 2:

We can find points on the graph of y = log3 x by letting x = 3k, where k is a real number,

y = log3 x = log3 3k = k

Using this formula, we obtain the points on the graph listed in the following table.

cont’d

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This gives us the same points obtained using the firstmethod.

Method 3:

We can sketch the graph of y = log3 x by sketching the graph of the equivalent exponential form x = 3y.

cont’d

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Logarithms with base 10 are called common logarithms. The symbol log x is used as an abbreviation for log10 x, just as is used as an abbreviation for .

The natural exponential function is given by f (x) = ex.

The logarithmic function with base e is called the natural logarithmic function.

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The symbol ln x (read “ell-en of x”) is an abbreviation for loge x, and we refer to it as the natural logarithm of x.

Thus, the natural logarithmic function and the natural exponential function are inverse functions of each other.

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Example 10 – Solving a simple logarithmic equation

Find x if

(a) log x = 1.7959 (b) ln x = 4.7

Solution:

(a) Changing log x = 1.7959 to its equivalent exponential form gives us

x = 101.7959.

Evaluating the last expression to three-decimal-place accuracy yields

x 62.503.

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(b) Changing ln x = 4.7 to its equivalent exponential form gives us

x = e4.7 109.95.

cont’d

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The following chart lists common and natural logarithmic forms

The last property for natural logarithms allows us to write the number a as eln a, so the exponential function f (x) = ax

can be written as f (x) = (eln a)x or as f (x) = ex ln a. Many calculators compute an exponential regression model of the form y = abx.

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If an exponential model with base e is desired, we can write the model

y = abx as y = aex ln b.

Illustration: Converting to Base e Expressions

• 3xis equivalent to ex ln 3

• x3 is equivalent to e3 ln x

• 4 2x is equivalent to 4 ex ln 2

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Figure 9 shows four logarithm graphs with base a > 1. Note that for x > 1, as the base of the logarithm increases, the graphs increase more slowly (they are more horizontal).

Figure 9

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This makes sense when we consider the graphs of the inverses of these functions: y = 2x, y = ex, y = 3x, and y = 10x.

Here, for x > 0, as the base of the exponential increases, the graphs increase faster (they are more vertical).

The next example illustrate application of common and natural logarithms.

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Example 11 – The Richter scale

On the Richter scale, the magnitude R of an earthquake of intensity I is given by

where I0 is a certain minimum intensity.

(a) If the intensity of an earthquake is 1000I0, find R.

(b) Express I in terms of R and I0.

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(a)

= log 1000

= log 103

= 3

given

let I = 1000I0

cancel l0

1000 = 103

log 10x = x for every x

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From this result we see that a tenfold increase in intensity results in an increase of 1 in magnitude (if 1000 were changed to 10,000, then 3 would change to 4).

(b) given

change to exponential form

multiply by l0

cont’d

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