Instruction: Solving Exponential Equations without Logarithms

This lecture uses a four-step process to solve exponential equations:

1 Isolate the base. 2 Write both sides of the equation as exponential expressions with like bases. 3 Set the exponents equal to each other. 4 Solve for the unknown.

In short, the strategy involves transforming the equation to the form ( ) ( )f x g xb b= then solving

( ) ( )f x g x= . Consider the equation ( ) 13281 12 =−+x .

Following the rubric, the base should be isolated. In other words, transform the equation algebraically so that the expression 2 12 x+ stands alone on the left side of the equation as shown below.

( )

( )

( )

( )

2 1

2 1

2 1

2 1

1 2 3 181 2 1 38

18 2 48

2 32

x

x

x

x

+

+

+

+

− =

= +

⎛ ⎞⋅ =⎜ ⎟⎝ ⎠

=

Now that the base is isolated, the second step of the rubric requires that the right side of the equation be written as an exponential expression with the same base as the left side (in this case, the base is two).

( )( ) 512

12

22

322

=

=+

+

x

x

The substitution above can be performed because 32 = 25. Now, the third step of the rubric urges setting the two exponents, 2x + 1 and 5, equal to one another. Logic maintains that the exponents must be equal if the bases are equal; thus,

512 =+x . Finally, the fourth step of the rubric requires solving the resulting equation as below.

2 1 52 4

2

xx

x

+ ===

This strategy only works for equations that can be transformed into the form ( ) ( )f x g xb b= . Later lectures will discuss exponential equations that require different solution strategies.

149

Instruction: The Irrational Number e

The quantity denoted by the symbol e has special significance in mathematics. It is an irrational number that is the base of natural logarithms, and it is often seen in real world problems that involve natural exponential growth or decay. The approximated value of e is 2.7182818284 . . . Students can replicate this value by using the e^ key on a graphing calculator and entering 1 as the exponent. Often a 2nd key must be used to generate e^. The housetop symbol, ^ , represents the fact that e will be raised to some power. Since e is often involved in exponential growth or decay problems, it is usually raised to some power when used for calculations; thus, the calculator anticipates the need to apply an exponent to e. Raising e to the first power replicates the approximated value of e. Students can also generate approximations of e by substituting extremely large numbers for m into the following expression:

This expression approaches e as the variable m approaches infinity. Functions of the form y = aekx or y = ae–kx where a and k are real number non-zero constants are often used in science problems. The following properties of exponents can be used to simplify expressions with e.

( ) ;;;1; 01 srsrsrsr aaaaaaaa +⋅ ==== srs

r

aaa −= ; and r

r

aa 1

=−

For example, consider( )

9 6 3

2

x x x

x x

e e e

e e

+ + .

( )( )

( )

3 9 3 6 3 3 39 6 3

2 2

3 6 3 0

2

3

=

=

x x x x x x xx x x

x xx x

x x x

x x

x

e e e ee e ee ee e

e e e e

ee

− − −

⋅

+

+ ++ +=

+ +

( )6 3

3

1x x

x

e e

e

+ +

6 3 1x xe e= + +

m

m⎟⎠⎞

⎜⎝⎛ +

11

150

Instruction: Graphing Exponential Functions

This lecture employs a step-by-step process for graphing exponential functions of the form ( )( ) kxy x a b= . The process applies for functions of this form even if the functions is translated.

For example, this discussion will consider 42)( 1 −= −xxk .

1. Identify the horizontal asymptote, usually y = 0. If the graph has a shift up or down c units, then the equation y = c describes the horizontal asymptote. Graph the horizontal asymptote accordingly. Since the asymptote is not actually part of the graph, use a dotted line to indicate its position.

2. Find the y-intercept by evaluating the function when the independent variable (the x-value) is zero:

27)0(

28

21)0(

421)0(

42)0(42)0(42)(

1

10

1

−=

−=

−=

−=

−=

−=

−

−

−

k

k

k

kk

xk x

•

151

3. If the function has a shift down, find the x-intercept. As discussed above, finding the x-intercept requires solving the exponential equation, which can require logarithms. For k(x), however, logarithms are not needed, which is sometimes the case as seen here.

3212242

042

21

1

1

==−=

=

=−

−

−

−

xx

x

x

x

Thus, solving the equation finds the x-intercept: .0)3( =k 4. Plug a few x-values into the function to find more values of the function:

x k(x) –1 –3.75 1 –3 4 4

Plot these points and sketch the curve remembering that exponential functions continually increase or continually decrease.

•

•

152

Instruction: Solving Exponential Equations without Logarithms

Example 1 Solving an Exponential Equations of the Form ( ) ( )f x g xb = b

Isolate the base.

( )( )( )

3

3

3

3

7 2 3 3587

7 2 3587 3

7 2 35847 7

2 512

x

x

x

x

+

+

+

+

+ =

= −

=

=

Recognize 512 as the ninth power of two.

3 92 2x+ = Note that the two exponents must be equal.

3 9If 2 2 , then 3 9x x+ = + = . Solve the linear equation.

3 99 36

xxx

+ == −=

Solve the equation for x: ( )37 2 3 3587x+ + = .

153

Example 2 Solving an Exponential Equations of the Form ( ) ( )f x g xb = b

Note that 125 is the third power of 5.

( )35 12

55

5

x

xx

− =

Recall that ( )sr r sa a ⋅= .

35 12 55

5

xx

x− =

Recall that r

r ss

a aa

−= for all non-zero values of a.

5 12 3

5 12 2

5 55 5

x x x

x x

− −

−

=

=

Note that the two exponents must be equal.

5 12 2If 5 5 , then 5 12 2x x x x− = − = . Solve the linear equation.

5 12 25 2 123 123 123 3

4

x xx xxx

x

− =− ==

=

=

Solve the equation for x: 5 12 12555

xx

x− = .

154

Instruction: The Irrational Number e

Example 1 The Irrational Number e

Recall the exponent property of real numbers, r s r sa a a +⋅ = : ( )17 1717 17 0 1e e e e+ −−⋅ = = = .

Example 2

The Irrational Number e

Recall the exponent property of non-zero real numbers, r

r ss

a aa

−= : 8

8 7 17 2.718e e e e

e−= = = ≈ .

Example 3

The Irrational Number e

Recall the exponent property of real numbers, ( )tr s r t s ta b a b⋅ ⋅= .

( )1 1 22 2 225 25

5

e ee

=

12⋅ 1

2255 5

ee e

=

Reduce appropriately and recall the exponent property 1

rra a= .

1225 e

5 e25 5 15 5

= = =

Evaluate 17 17e e−⋅ without using a calculator. Round the answer to the nearest thousandths.

Evaluate 8

7

ee

without using a calculator. Round the answer to the nearest thousandths.

Evaluate ( )

12 225

5

e

e without using a calculator.

155

Example 4 The Irrational Number e

Recall the exponent properties of real numbers, r s r sa a a +⋅ = and ( )sr r sa a ⋅= .

( )35 5 3 15

7 6 7 6 13

e e ee e e e

⋅

+= =⋅

Recall the exponent property of non-zero real numbers, r

r ss

a aa

−= .

15

15 13 213

e e ee

−= =

Use the calculator xe key to evaluate 2e . Round to the nearest thousandth.

2 7.389e ≈

Example 5 The Irrational Number e

Recall that division with fractions is multiplication of the dividend by the reciprocal of the divisor.

5

5 5 56

5 6 3

3

ee e ee

e e ee

= ÷ =3

6 5

ee e

⋅3

6

ee

= .

Recall the exponent property of non-zero real numbers, 1r

s s r

aa a −= .

3

6 6 3 3

1 1ee e e−= =

Evaluate ( )35

7 6

e

e e⋅. Round the answer to the nearest thousandth.

Simplify

5

6

5

3

eeee

. Leave all answers exact.

156

Example 6 The Irrational Number e

Recall the exponent property of real numbers, r s r sa a a +⋅ = and simplify the denominator.

2x x x x xe e e e+⋅ = = . Factor the greatest common factor in the numerator.

( ) ( )2 5 2 4 2 2 2 2 3 2 05 4 2

2 2 2

2 22x x x x x x x x x xx x x

x x x

e e e e e e e ee e ee e e

− − −− + − +− += =

Simplify and reduce.

2xe ( )3 2 0

2

2x x

x

e e e

e

− +3 2 3 22 1 2x x x xe e e e= − + ⋅ = − +

Simplify 5 4 22x x x

x x

e e ee e− +

⋅.

157

Instruction: Graphing Exponential Functions

Example 1 Graphing Exponential Functions

The initial amount (the coefficient to the base) is 1. The y-intercept is (0,1). The asymptotic value is zero. The common ratio is 2. Since the common ratio is greater than one, the function increases and approaches zero as x approaches negative infinity, making the horizontal asymptote y = 0 (the x-axis).

Graph ( ) 2xy x = . Label intercepts and asymptotes. Show proper behavior.

158

Example 2 Graphing Exponential Functions

The initial amount (the coefficient to the base) is 3. The y-intercept is (0,3). The asymptotic value is zero. The common ratio is 1.5. Since the common ratio is greater than one, the function increases and approaches zero as x approaches negative infinity, making the horizontal asymptote y = 0 (the x-axis).

Graph ( ) ( )3 1.5 xq x = . Label intercepts and asymptotes. Show proper behavior.

159

Example 3 Graphing Exponential Functions

The initial amount (the coefficient to the base) is 10. The y-intercept is (0,10). The asymptotic value is zero. The common ratio is 0.5. Since the common ratio is less than one, the function decreases and approaches zero as x approaches infinity, making the horizontal asymptote y = 0 (the x-axis).

Graph ( ) ( )10 0.5 xq x = . Label intercepts and asymptotes. Show proper behavior.

160

Example 4 Graphing Exponential Functions

The initial amount (the coefficient to the base) is 1. The asymptotic value is negative two. Since 1 + –2 = –1, the y-intercept is (0, –1). The common ratio is 1 4 . Since the common ratio is less than one, the function decreases and approaches negative two (the asymptotic value) as x approaches infinity, making the horizontal asymptote y = –2. Setting the equation equal to zero and solving yields the x-value of the x-intercept.

( )( )

( )

( )

12

1 2 04

4 2

4 4

4 41 21 2

1 2,0

x

x

x

x

xx

−

−

−

⎛ ⎞ − =⎜ ⎟⎝ ⎠

=

=

=

− == −∴

−

Graph ( ) 1 24

x

g x ⎛ ⎞= −⎜ ⎟⎝ ⎠

. Label intercepts and asymptotes. Show proper behavior.

161

Suggested Homework from Blitzer Section 4.1: #27, #28, #32, #37, #39, #41, #43, #47, #51

Application Exercise

In the latter half of the 20th century, American sociologists used ( ) 0.00586.9 tp t e−= to model the percent of households that were married households between 1970 and 2000 where t represents years since 1970. According to the model, what percent of households were married households in 1990?

162

ANSWERS

#1 x = ½ #3 x = –4 #5 x = –3 #7 x = 5/4 #9 –e

x – 1

#11 #13 #15

0,0 0,1

250630,

HA: 500y

Practice Problems

Solve the following equations.

#1 255 12 x #2 14 927 xx

#3 813

1

x

#4

x

x

8

275.1 1 Hint:

8

27

2

33

#5 32

12 12 x #6

x

x

2

162

#7 3216 x #8 x1.0000,1

Simplify the expressions below using properties of exponents.

#9 2

232 2

x

xxx

e

eee #10

x

xxx

e

eee 11 2

Graph the following functions. Label the intercepts.

#11 12)( xxt #12 ( ) 2xf x e

#13 xxy 2)( #14 12

( ) 3 12x

h x

For fun, create a table of values and graph the following special type of exponential function

called a logistic growth function or sigmoidal curve.

#15 0.33

500( )

1 125 xQ x

e

163