Copyright © 2011 Pearson Education, Inc. Slide 5.5-1 5.5 Exponential and Logarithmic Equations and...

Copyright © 2011 Pearson Education, Inc. Slide 5.5-1

5.5 Exponential and Logarithmic Equations and Inequalities

Properties of Logarithmic and Exponential Functions

For b > 0 and b 1,

1. if and only if x = y.

2. If x > 0 and y > 0, then logb x = logb y if and only if x = y.

yx bb


5.5 Exponential and Logarithmic Equations and Inequalities

• Type I Exponential Equations – Solved in Section 5.2

– Easily written as powers of same base

i.e. 125x = 5x

• Type 2 Exponential Equations – Cannot be easily written as powers of same base

i.e 7x = 12

– General strategy: take the logarithm of both sides and apply the power rule to eliminate variable exponents.


Example Solve 7x = 12.

Solution

5.5 Type 2 Exponential Equations

277.17ln

12ln12ln7ln

12ln7ln

127

xx

x

x


5.5 Solving a Type 2 Exponential Inequality

Example Solve 7x < 12.

Solution From the previous example, 7x = 12 when

x 1.277. Using the graph below, y1 = 7x is below the graph y2 = 12 for all x-values less than 1.277.

The solution set is (–,1.277).


5.5 Solving a Type 2 Exponential Equation

Example Solve

Solution

.32 413 xx

3log2log32log3log4

2log3log4)3log2log3(

2log3log43log2log3

3log3log42log2log3

3log)4(2log)13(

3log2log

32413

413

x

x

xx

xx

xx

xx

xx

Take logarithms of both sides.

Apply the power rule.

Distribute.

Get all x-terms on one side.

Factor out x and solve.


5.5 Solving a Logarithmic Equation of the Type log x = log y

Example Solve

Analytic Solution The domain must satisfy x + 6 > 0, x + 2 > 0, and x > 0. The intersection of these is (0,).

.log)2(log)6(log 333 xxx

xxx

xxx

xxx

26

log26

log

log)2(log)6(log

33

333

Quotient property of logarithms

log x = log y x = y



Since the domain of the original equation was (0,),

x = –3 cannot be a solution. The solution set is {2}.

2or3

)2)(3(0

60

26

)2(6

2

2

xx

xx

xx

xxx

xxx Multiply by x + 2.

Solve the quadratic equation.



Graphing Calculator Solution

The point of intersection is at x = 2. Notice that the graphs do not intersect at x = –3, thus supporting our conclusion that –3 is an extraneous solution.


5.5 Solving a Logarithmic Equation of the Type log x = k

Example Solve

Solution

Since it is not in the domain and must be discarded, giving the solution set

.1)1log()23log( xx

61451

0123

1023

10)1)(23(

1)1)(23(log

1)1log()23log(

2

2

1

x

xx

xx

xx

xx

xx

Write in exponential form.

,161451

.161451


5.5 Solving Equations Involving both Exponentials and Logarithms

Example Solve

Solution The domain is (0,).

.161ln2 xe

4

4

161

161

161

22

2

rulePower ln

ln2

ln

2

x

x

x

e

e

ue

x

x

u

– 4 is not valid since – 4 < 0, and x > 0.


5.5 Solving Exponential and Logarithmic Equations

An exponential or logarithmic equation can be solved by changing the equation into one of the following forms, where a and b are real numbers, a > 0, and a

1:

• a f(x) = bSolve by taking the logarithm of each side.

1. loga f (x) = loga g (x) Solve f (x) = g (x) analytically.

3. loga f (x) = bSolve by changing to exponential form f (x) = ab.


5.5 Solving a Logarithmic Formula from Biology

ExampleThe formula gives the number of species in a sample, where n is the number of individuals in the sample, and a is a constant indicating diversity. Solve for n.

Solution Isolate the logarithm and change to exponential form.

anaS 1ln

)1(

1

1ln

aS

aS

ean

an

e

an

aS

Copyright © 2011 Pearson Education, Inc. Slide 5.5-1 5.5 Exponential and Logarithmic Equations and...

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