67
Example. Determine the inverse of the given function (if it exists).
p
3
f (x) =
g(x) = 5 x + 2
x 1
We know want to look at two di↵erent types of functions, called logarithmic functions and
exponential functions. It will turn out that these two types of functions are inverses of each
other, i.e. the inverse of a logarithmic function will be an exponential function and the inverse
of an exponential function will be a logarithmic function.
Exponential Functions (Section 5.2)
Definition. The function
x is a real number, a > 0, and a 6= 1, is called the exponential function with base a.
Example. Consider the exponential functions
f (x) = 2
x
and
Find the following output values.
✓ ◆x
1
g(x) =
2
and
h(x) = ex .
f ( 2) =
g( 2) =
h( 2) =
f (0) =
g(0) =
h(0) =
f (2) =
g(2) =
h(2) =
68
✓ ◆x
1
Let’s draw the graphs of f (x) = 2 and f (x) =
:
2
x
9
8
7
6
5
4
3
2
-9-8-7-6-5-4-3-2-1 1
-1 1 2 3 4 5 6 7 8 9
-2
-3
-4
-5
-6
-7
-8
-9
9
8
7
6
5
4
3
2
-9-8-7-6-5-4-3-2-1 1
-1 1 2 3 4 5 6 7 8 9
-2
-3
-4
-5
-6
-7
-8
-9
✓ ◆x
1
Now, let’s draw the graphs of f (x) = 3x and f (x) =
:
3
9
8
7
6
5
4
3
2
-9-8-7-6-5-4-3-2-1 1
-1 1 2 3 4 5 6 7 8 9
-2
-3
-4
-5
-6
-7
-8
-9
So graphs of exponential functions look like:
9
8
7
6
5
4
3
2
-9-8-7-6-5-4-3-2-1 1
-1 1 2 3 4 5 6 7 8 9
-2
-3
-4
-5
-6
-7
-8
-9
-1
1
69
Logarithms (Section 5.3)
What does loga x mean????
Example. Calculate the following.
log2 8
log5 1
log3 81
✓ ◆
1
log2
8
log1 00.001
log4 16
Definition. The function
is the logarithmic function having base a, where a > 0 is a real number.
Two bases that occur often have special notations:
70
Example. Compute the following. Round your answers to two decimal places.
log 7
ln 2
log 30
ln 20
Your calculator has buttons for ln x and log x, but how do you calculate other base logarithms?
Example. Compute the following. Round your answers to two decimal places.
log3 15
log2 6
log6 30
You can convert between logarithmic and exponential equations using the following:
71
Example. Convert the following logarithmic equations to exponential equations.
log5 5 = 1
log 7 = 0.845
ln 40 = x
Example. Convert the following exponential equations to logarithmic equations.
1
e3 = t
5 3=
103 = 1000
125
Graphs of Inverse Functions (Section 5.1)
Let’s compare the graphs of two inverse functions. Graph f (x) = 2x
on the same coordinate axes:
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9
3 and f
1
(x) =
x+3
2
72
Let’s try another pair of inverse functions. Graph f (x) = x3 + 2 and f
same coordinate axes:
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
Since f (x) = ax and f
loga (x).
1
1
(x) =
p
3
x
2 on the
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9
(x) = loga (x), we can use the graph of f (x) = ax to graph f
1
(x) =
Example. Sketch the graph of each logarithmic equation.
y = log2 x
y = log 1 x
2
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9
73
Application to Compound Interest (Section 5.2)
If P dollars are invested at an interest rate r (written as a decimal), compounded n times per
year, then the amount A of money in the account after t years is given by:
⇣
r ⌘nt
A=P 1+
n
Example. If $500 is invested at a rate of 1.25%, compounded monthly, how much money will
be in the account after 5 years?
Example. If $1200 is invested at a rate of 2.5%, compounded semiannually, how much money
will be in the account after 2 years?
As the number of times you compound interest per year increases, the amount in the account
after t years also increases. Consider investing $100 at a rate of 2% for a time period of 5 years.
So
✓
◆5n
.02
A = 100 1 +
n
74
If interest is compounded continuously, then the amount A in the account after t years is:
A = P ert
Example. If $500 is invested at a rate of 1.25%, compounded continuously, how much money
will be in the account after 5 years?
Example. If $1200 is invested at a rate of 2.5%, compoundedcontinuously, how much money
will be in the account after 2 years?
Solving Exponential and Logarithmic Equations (Section 5.5)
Logarithm and exponential functions are inverses, so:
loga (ax ) = x
Example. Solve the following equations:
2x+3 = 5
and
aloga (x) = x
log5 (3x
1) = 12
75
Example. Solve the following equations:
54x
7
= 125
2x = 40
Properties of logarithms:
• loga M + loga N = loga M N
• loga M
loga N = loga
• p loga M = loga M p
M
N
log5 (8
7x) = 3
ln x =
2
76
So, how do we solve logarithmic equations like
log8 (x + 1)
Example. Solve the following equations:
log2 (x + 1) + log2 (x 1) = 3
log8 x = 2
log x + log(x + 4) = log 12
77
One more example . . . solve
ln(x + 8) + ln(x
1) = 2 ln x
Let’s take a look back at solving exponential equations.
Example. Solve the following equations:
1000e0.09t = 5000
250
(1.87)x = 0
78
Example. Suppose that $1,500 is invested at an interest rate of 1.75%, compounded quarterly.
How long would it take for the amount of money in the account to become $2,000?
Example. Suppose that $5,000 is invested at an interest rate of 5.4%, compounded continuously.
How long would it take for the amount of money to double?