Chapter Nineteen — Exponential Functions
The Humongous Book of Algebra Problems
423
Note: Problems 19.6–19.9 refer to the function g(x) = e
x
.
19.9 Graph .
To transform the function g(x) = e
x
into , multiply g(x) by –1 (which
reflects the graph across the x-axis) and subtract 4 from the input (which shifts
the graph four units to the right). The graph of ˆg(x) is presented in Figure
19-5.
Figure 19-5: The graph of is the graph of g(x) = e
x
reflected across the
x-axis and shifted four units to the right.
Composing Exponential and Logarithmic Functions
They cancel each other out
Note: Problems 19.10–19.11 prove that h(x) = log
3
x and k(x) = 3
x
are inverse functions.
19.10 Demonstrate that h(k(x)) = x.
Substitute k(x) = 3
x
into h(x) = log
3
x.
If f(g(x)) =
g(f(x)) = x, then
f(x) and g(x) are
inverse functions.
Problem 19.10 proves
that f(g(x)) = x, and
Problem 19.11 proves
that g(f(x)) = x.
Chapter Nineteen — Exponential Functions
The Humongous Book of Algebra Problems
424
Express the logarithmic equation as an exponential equation.
3
h(k(x))
= 3
x
If equivalent exponential expressions have equal bases, then the exponents
must be equal as well.
h(k(x)) = x
Note: Problems 19.10–19.11 prove that h(x) = log
3
x and k(x) = 3
x
are inverse functions.
19.11 Demonstrate that k(h(x)) = x.
Substitute h(x) = log
3
x into k(x) = 3
x
.
Express the exponential equation as a logarithmic equation.
If two equivalent logarithms have the same base, then the arguments of those
logarithms must also be equal.
k(h(x)) = x
19.12 Simplify the expression: log
6
6
5x
.
Substituting 6
5x
(an exponential function with base 6) into its inverse (a
logarithmic function with base 6) produces an expression equal to the
exponent: log
b
b
f(x)
= f(x).
log
6
6
5x
= 5x
19.13 Simplify the expression: .
Substituting a logarithmic function with base 2 into an exponential function
with base 2 produces an expression equal to the argument of the logarithm:
.
Note: Problems 19.14–19.15 refer to the expression ln (9e
x
).
19.14 Apply a logarithmic property to expand the logarithmic expression.
According to Problem 18.34, the logarithm of a product is equal to the sum of
the logarithms of its factors: log
a
(xy) = log
a
x + log
a
y.
ln
(9e
x
) = ln
9 + ln
e
x
If you plug
an exponential
function into a log
with the same base,
everything disappears
except for the
exponent.
If the power
of an exponential
function is a log
with the same base,
everything disappears
except for whatever’s
inside the logarithm, in
this case x
˙
+ 1.
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:
.
..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.
Reset