Chapter Nineteen — Exponential Functions

The Humongous Book of Algebra Problems

426

The natural exponential function and the natural logarithmic function are

inverses, so composing them eliminates them from the equation.

= y

2

Therefore, .

Exponential and Logarithmic Equations

Cancel logs with exponentials and vice versa

19.19 Solve the equation for x: 25

x

= 125.

Express both sides of the equation as a power of ﬁve: 25 = 5

2

and 125 = 5

3

.

Two equivalent exponential expressions with equal bases must also have

exponents that are equal.

2x = 3

Solve for x.

19.20 Solve the equation for x: 8

x

= 128.

Express both sides of the equation as a power of two: 8 = 2

3

and 128 = 2

7

.

Chapter Nineteen — Exponential Functions

The Humongous Book of Algebra Problems

427

19.21 Solve the equation for x: 3

x

= 19.

To eliminate the exponential function 3

x

—thereby isolating x and solving the

equation—apply its inverse function (log

3

x) to both sides of the equation.

19.22 Solve the equation for x: 2 – 5e

x

= –17.

Isolate e

x

on the left side of the equation.

Take the natural logarithm of both sides of the equation to eliminate the

natural exponential function and solve for x.

19.23 Solve the equation for x: e

2x

– 6e

x

= 0.

Express e

2x

as (e

x

)

2

.

(e

x

)

2

– 6e

x

= 0

Factor e

x

out of the expression left of the equal sign.

e

x

(e

x

– 6) = 0

Apply the zero product property, setting both factors equal to zero.

e

x

= 0 or e

x

– 6 = 0

Solve the equations for x.

Note that the natural logarithm function is not deﬁned for x = 0, so x = ln 0 is

not a valid solution. The only solution to the equation is x = ln

6.

You can’t do this

problem the way

you did Problems 19.19

and 19.20 because you

can’t write 3 and 19

using a common base.

That means you have

to remove x from the

exponent (using log

3

)

and solve for it.

Even though

it looks ugly, the

exact solution is

log

3

19. Don’t use a

calculator and/or the

change of base formula

to give an approximate

decimal answer unless a

problem specically

tells you to.

If you multi-

ply the powers

of (e

x

)

2

, you get e

2x

.

Why write e

2x

as (e

x

)

2

?

When you do, it’s a little

easier to factor e

x

out

of the expression in the

next step—you factor

one e

x

out of

(e

x

)

2

= (e

x

)(e

x

), leaving

one behind.

Look at the

graphs of the log

functions in Problems 18.12

and 18.15. They don’t

intersect the y-axis (x = 0)

or any vertical line left

of the y-axis.

Chapter Nineteen — Exponential Functions

The Humongous Book of Algebra Problems

428

19.24 Solve the equation for x: e

2x

– 9e

x

= –14.

Like Problem 19.23, this equation can be solved by factoring. Therefore, like the

preceding problem, one side of the equation must equal zero in order to apply

the zero product property.

e

2x

– 9e

x

+ 14 = 0

Express e

2x

as (e

x

)

2

and solve the equation by factoring.

The solution to the equation is x = ln

2 or x = ln

7.

19.25 Solve the equation for x: ln

x = 4.

In this equation, x is the argument of a natural logarithm—a logarithm with

base e. To eliminate the logarithmic expression, exponentiate both sides of the

equation with base e.

e

ln x

= e

4

Because e

x

and ln

x are inverses, composing the functions eliminates both of

them from the equation.

x = e

4

19.26 Solve the equation for x: log

3

x = 2.

To eliminate the logarithmic expression, and thereby isolate x on the left side

of the equation, exponentiate both sides of the equation with the base of the

logarithm, 3.

19.27 Solve the equation for x: ln

(3x + 7) = 13.

To eliminate the natural logarithm, exponentiate both sides of the equation

with base e.

Add 14 to

both sides of the

equation.

“Exponentiate

with base e”

means turn

something into an

exponent with base e.

When you exponentiate

to solve a log equation,

isolate the log

containing x on one side

of the equal sign and

then make both sides

exponents that

have the same

base as the log.

The exact

answer is e

4

. Don’t

use a calculator

to approximate the

value of an answer

containing e unless

you’re specically

instructed to

do so.

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