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The Humongous Book of Algebra Problems by W. Michael Kelley

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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 cant do this
problem the way
you did Problems 19.19
and 19.20 because you
cant 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. Dont use a
calculator and/or the
change of base formula
to give an approximate
decimal answer unless a
problem specically
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 dont
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
. Dont
use a calculator
to approximate the
value of an answer
containing e unless
youre specically
instructed to
do so.

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