Chapter Nineteen — Exponential Functions
The Humongous Book of Algebra Problems
425
Note: Problems 19.14–19.15 refer to the expression ln (9e
x
).
19.15 Simplify the expanded logarithm expression generated by Problem 19.14.
The natural exponential function e
x
and the natural logarithm function ln
x
have the same base, e. Therefore, substituting e
x
into ln
x results in an expression
equal to the exponent: ln
e
x
= x.
f(x) = ln
9 + ln
e
x
= ln
9 + x
Therefore, ln
(9e
x
) = x + ln
9.
19.16 Apply a logarithmic property to simplify the expression: .
According to Problem 18.36, the logarithm of a quotient is equal to the
difference of the logarithms of the dividend and divisor: .
Simplify the expression log
10
x – 4
.
= x – 4 – log
3
19.17 Apply a logarithmic property to simplify the expression: .
Express the quotient as the difference of the logarithm of the numerator
and the logarithm of the denominator.
Substituting 4
x
into its inverse function, a logarithm with base 4, eliminates
both functions: log
4
4
x
= x.
19.18 Apply a logarithmic property to simplify the expression: .
According to Problem 18.38, the logarithm of a quantity raised to an exponent
n is equal to the product of n and the logarithm of the base: n · log
a
x = log
a
x
n
.
Rewrite the natural logarithm by transforming its coefficient (4) into the
exponent of the argument : . Express the square root
as rational exponent and simplify: .
Common
logs (like
log
10
x – 4
) have
an implied base of
10. In other words,
log
10
x – 4
= log
10
10
x – 4
.
You have an expo-
nential function with
base 10 plugged into a
log of base 10. They
cancel each other out,
leaving behind the
exponent
x – 4.
Wonder
where this
number came
from? Set
log
4
2 = c and
create the
exponential equation
4
c
= 2. Write
both sides of the
equation as powers
of 2 and solve for c:
To raise y
1/2
to
the fourth power,
multiply the powers:
.