163
LOGARITHMS
Simplifying Logarithms…………………………………………166
Properties of Logarithms…….…………………………….……167
Expanding Logarithmic Expressions…….……………….…….167
Condensing Logarithmic Expressions…….....………..………..169
Practice Using Properties of Logarithms…………………..…..170
Base Change Formula……………………………………..…….172
Solving Logarithmic Equations…….……………………...……173
Solving Exponential Equations…………………………………175
Finding the Domain of a Logarithmic Function….……..……..177
Finding the Vertical Asymptote of a Logarithmic Function….178
Graphing Logarithmic Functions.……………………….……..179
Finding the Inverse of a Function…….…………………..…….184
Interest Formulas……………………………………….……….187
Word Problems………….……………………………...………..191
164
Objectives
The following is a list of objectives for this section of the workbook.
By the time the student is finished with this section of the workbook, he/she should be able
to…
• Evaluate a simple logarithm without the aid of a calculator.
• Express a logarithmic statement is exponential form.
• Express a statement in exponential form in logarithmic form.
• Expand a logarithmic expression as the sum or difference of logarithms using
the properties of logs.
• Condense the sum or difference of logarithms into a single logarithmic
expression.
• Evaluate logarithms using the base change formula.
• Solve logarithmic equations.
• Evaluate the solution to logarithmic equations to find extraneous roots.
• Solve equations with variables in the exponents.
• Find the range and domain of logarithmic functions.
• Graph a logarithmic function using a table.
• Find the inverse of a function.
• Verify two functions are inverses of each other.
• Identify a one-to-one function.
• Use the compound interest formulas.
Math Standards Addressed
The following state standards are addressed in this section of the workbook.
Algebra II
11.0 Students prove simple laws of logarithms.
11.1 Students understand the inverse relationship between exponents and logarithms and use
this relationship to solve problems involving logarithms and exponents.
11.2 Students judge the validity of an argument according to whether the properties of real
numbers, exponents, and logarithms have been applied correctly at each step.
13.0 Students use the definition of logarithms to translate between logarithms in any base.
14.0 Students understand and use the properties of logarithms to simplify logarithmic numeric expressions and to
identify their approximate values.
15.0 Students determine whether a specific algebraic statement involving rational expressions, radical expressions,
or logarithmic or exponential functions is sometimes true, always true, or never true.
24.0 Students solve problems involving functional concepts, such as composition, defining the inverse function and
performing arithmetic operations on functions.
165
Logarithms are similar to radicals in that knowing what the question is asking makes the
problem easier. Although this is a topic that is completely new to Algebra II students,
Logarithms are simple. For example, the question
3
log 27
=
is asking “To what power do you
raise 3 to get 27?” In this particular problem, 3 is the base of the logarithm. When reading
the logarithm, it is read “Log base 3 of 27 is…”
Properties of Simple Logarithms
log
log 1 0
log 1
log ( )
log log
a
a
a
x
x
a
a a
a
a x and a x inverse property
If x y then x y
=
=
= =
= =
Properties of Natural Logarithms
ln
ln1 0
ln 1
ln ( )
ln ln
x x
e
If x y then x y
=
=
= =
= =
A standard logarithm can have any positive number as its base except 1, whereas a natural log
is always base
e
. Since the natural log is always base
e
, it will be necessary to use a
calculator to evaluate natural logs unless one of the first three examples of the properties of
natural logs is used. For anything such as
ln2
=
, a calculator must be used.
When dealing with logarithms, switching between exponential and Logarithmic form is often
necessary.
Logarithmic form Exponential Form
log
a
b c
=
c
a b
=
Write each of the following in exponential form.
A)
4
log 16 2
=
B)
9
1
log 3
2
=
C)
9
3
log 27
2
=
E)
4
1
log 2
16
= −
Write each of the following in logarithmic form.
A)
4
3 81
=
B)
1 4
16 2
=
C)
1 2
1
36
6
−
=
D)
5 4
16 32
=
166
Simplifying Logarithms
Evaluate each of the following logarithms without the use of a calculator.
A)
3
log 81
=
B)
4
1
log
2
=
C)
12
log 144
=
D)
6
1
log
36
=
E)
2
3
9
log
4
=
F)
0.25
log 4
=
G)
3
log 3
− =
H)
8
log 4
=
I)
81
1
log
27
=
J)
1
16
log 32
=
K)
4
log 0
=
L)
10
log 1
=
M)
4
1
log
8
=
N)
27
1
log
3
=
O)
9
log 3
=
P)
3
6
log 6
x
=
Q)
36
1
log
6
=
R)
128
log 2
=
S)
1
4
log 16
=
T)
2
log
x
z
z
=
U)
12
lne
=
V)
3
log 5
3
=
W)
ln1
=
X)
ln4x
e
=
Y)
2
log 16 2
=
Z)
5
3
log 9
=
a)
3
3
log 9 3
=
b)
5
3
1
log
25
=
c)
3
5
6
36
log
25
=
d)
2
ln5
x
e
=
167
Properties of Logarithms
The following properties serve to expand or condense a logarithm or logarithmic expression so
it can be worked with.
Properties of logarithms Example
log log log
log log log
log log
a a a
a a a
n
a a
mn m n
m
m n
n
m n m
= +
= −
=
( )
( ) ( )
4 4 4
2 2 2
3
3 3
log 3 log 3 log
1
log log 1 log 5
5
log 2 1 3log 2 1
x x
x
x
x x
= +
+
= + −
+ = +
Properties of Natural Logarithms Example
ln ln ln
ln ln ln
ln ln
n
mn m n
m
m n
n
m n m
= +
= −
=
(
)
(
)
(
)
(
)
3
ln 1 5 ln 1 ln 5
ln ln ln2
2
ln7 3ln7
x x x x
x
x
+ − = + + −
= −
=
These properties are used backwards and forwards in order to expand or condense a
logarithmic expression. Therefore, these skills are needed in order to solve any equation
involving logarithms. Logarithms will also be dealt with in Calculus. If a student has a firm
grasp on these three simple properties, it will help greatly in Calculus.
Expanding Logarithmic Expressions
Write each of the following as the sum or difference of logarithms. In other words, expand
each logarithmic expression.
A)
3 2
2
5
3
log
x y
z
B)
2
3
3
log 5
xy
C)
( ) ( )
3 2
4
log 1 2
x x+ − D)
2
5
5
6
log
11
x
y z
168
E)
( )
3
5
2
3 2
log
1
x
x
+
−
F)
12
7
log
2
x
x
−
+
G)
3
log 12
a
x y
H)
5 3
3
3
2
5
log
x y
z
I)
2
3
4
ln
x
x
−
J)
5
ln3
x y
K)
( )
3
ln 4
x x
+
L)
( )
2
3
ln 3
x x
−
M)
3
5
ln
x y
z
N)
ln
2
x
x
−
169
Condensing Logarithmic Expressions
Rewrite each of the following logarithmic expressions using a single logarithm. Condense
each of the following to a single expression. Do not multiply out complex polynomials. Just
leave something like
( )
3
5
x +
alone.
A)
4 4 4
3log 5log 2log
x y z
− +
B)
1
2log log
2
x y
+
C)
1 1 2
log6 log log
3 3 3
x y
+ +
D)
3
3 3 3
3 1
log 16 log 2log
4 3
x y
− −
E)
5 5 5
3log 2log log 2
x y z
+ + +
F)
2 2
1 2
log log 3
3 3
x y
+ −
G)
(
)
(
)
(
)
3 3 3
log 2 log 2 log 4
x x x
+ + − − +
H)
( ) ( ) ( )
2 1 1
log 1 log 2 log 5
3 3 3
x x x
+ + − − +
I)
( ) ( )
1
2ln 3 ln ln 2 1
3
x x x
+ + − −
J)
(
)
(
)
(
)
ln 3 ln 2 5 2ln 1
x x x
+ − + + −
170
K)
( ) ( )
1
ln 3 2ln 1 3ln
2
x x x
+ + − −
L)
2
2ln3 6ln ln27
3
x+ −
M)
3
1 log
2
x
+
N)
(
)
(
)
ln 2 ln 2 ln 2
x x x− + + −
Practice Using Properties of Logarithms
Use the following information, to approximate the logarithm to 4 significant digits by using
the properties of logarithms.
log 2 0.3562, log 3 0.5646, log 5 0.8271
a a a
and≈ ≈ ≈
A)
6
log
5
a
B)
log 18
a
C)
log 100
a
D)
log 30
a
E)
log 3
a
F)
log 75
a
G)
4
log
9
a
H)
3
log 15
a
I)
2
log 54
a
171
Use the following information for letters J – R.
10 10 10
log 2 0.3010, log 3 0.4771, log 7 0.8451
and≈ ≈ ≈
J)
10
log 28
K)
10
log 8
L)
10
log 4.5
M)
10
log 98
N)
10
log 20
O)
10
log 210
P)
10
1
log
300
Q)
10
1
log
9
R)
10
log 42
172
Base Change Formula
Up to this point calculators have not been used to evaluate logarithms. Remember, the logs on
all calculators are base 10. If a calculator is used to evaluate a log with any other base, it will
not give the correct estimate unless a specific formula is used. The following base change
formula will be used to evaluate any log with a base other than 10.
log
log
log
a
b
b
a
=
The terms
log
a
and
log
b
will be evaluated with a base 10, which means a calculator can now
be used. Of course natural logs are always base
e
, so there is no need for a formula with
natural logs. Each calculator is different. Be aware of what keystrokes are needed for your
particular calculator, whether the log must be entered first, or the number first. Be very
careful about grouping symbols when entering these types of problems into the calculator.
Using a calculator, evaluate each of the following. Round all answers to three decimal
places.
A)
3
log 12
B)
6
log 17
C)
3
1
log
5
D)
4
log 8
E)
6
1
log
12
F)
(
)
7
log 35
−
G)
ln14
H)
ln3.26
I)
1
ln
2
J)
ln0
K)
ln
e
L)
ln6.2
Why can’t you take the log of a negative number?
173
Solving Logarithmic Equations
Here we will solve some logarithmic equations. There are a couple of ways to solve a
logarithmic equation. One method is to get all logs to the left side of the equal sign and
condense the problem so there is only one log. The next step would be to write the problem in
exponential form and solve.
Another method involves one of the properties of simple logarithms. In particular, the
property that states: “
log log
a a
If x y then x y
= =
.” If the equation can be manipulated to
resemble this property, take what is inside of each and set the two terms equal to each other.
This follows the same principal as exponential equations where the bases are identical. If an
exponential equation states
3 4 7
2 2
x
+
=
; it stands to reason, because of the equal sign, that
3 4 7
x
+ =
. The problem would then be solved from this point. With logarithmic equations,
the problem
(
)
4 4
log 8 5 log 12
x − =
can be written as
8 5 12
x
− =
.
Always check for extraneous roots!!!
Solve each of the following logarithmic equations.
(Round any solutions with decimals to three
decimal places)
A)
4
log 6 8
x
+ =
B)
3
2log 4 1
x
− =
C)
4
5 2log 4
x
+ =
D)
3 3
log log 2 5
x
− =
E)
(
)
log 2 1 0
x
− =
F)
(
)
log 4 8 2
x
+ =
G)
(
)
(
)
3 3 3
log 5 log 3 log 35
x x+ + + =
H)
(
)
(
)
log 3 log 4 log30
x x+ + − =
174
I)
3
log 3 2 2
x
− =
J)
(
)
5 5 5
log log 4 4log 2
x x+ + =
K)
log 625 4
x
=
L)
(
)
(
)
2 2 2
log 2 log 6 log 13
x x− + − =
M)
(
)
3 3
2log log 2 2
x x
− − =
N)
ln ln 2 0
x
− =
O)
(
)
ln 3 5 8
x
+ =
P)
(
)
ln 1 3.2
x − =
Q)
(
)
(
)
3 3
log 4 2 log 6
x x
− = +
R)
(
)
ln 3 2 4
x
− =
S)
7
ln 10
x
e
+
=
T)
( )
2
2
log 7 log 64
3
a a
x + =
U)
(
)
(
)
2 2
log 3 log 3 4
x x
+ + − =
175
Solving Exponential Equations
You will now be solving exponential equations. Logarithms allow you to solve equations with
variables in the exponent. We have dealt with these types of problems before, however, in the
previous examples the bases of the problem could be made to match making solving the
problem as simple as setting the two exponents equal to each other and solving for the
variable. When ever you come to a problem with the variable in the exponent, and you cannot
get the bases to match, you will need to solve the problem using logarithms. You will be using
all three properties of logarithms to simplify and eventually solve the problem. You can take
either the log or natural log of both sides, it makes no difference unless
e
is in the problem.
In that case, you must use natural logs to solve the equation.
Solve each of the following exponential equations. Round solutions to three decimal places.
A)
2
3 5
x
=
B)
4
6 300
x
=
C)
3 1 2
12 7
x+
=
D)
2 5
5 3
x+
=
E)
5 8 1
4 8
x x
+ −
=
F)
3 2 8
9 27
x x
+ +
=
G)
3 2 5 1
12 8
x x
− +
=
H)
6 5 2 1
5 6
x x
− +
=
176
I)
3
4 40
x
e
=
J)
1200 900
x
e =
K)
4
3 12
x
e
=
L)
2
3 2 0
x x
e e
− + =
M)
2
5 6 0
x x
e e
− + =
N)
2
2 24 0
x x
e e
+ − =
O)
2
2 10 0
x x
e e
+ − =
P)
2
5 7 5 10 0
x x
− ⋅ + =
Q)
2
3 5 3 14 0
x x
+ ⋅ − =
R)
( )
( )
8 4 6
1
log log log
3
x
=
S)
(
)
2 4
log log 1
x
=
T)
(
)
3 27
log log 1
x
= −
177
Finding the Domain of a Logarithmic Function
Since we can’t take the log of zero or the log of a negative number, we can find the domain of
a logarithmic function using a simple inequality. Set what is inside the log greater than zero
and solve. The result will give you the domain of the function. Consider the following.
( )
(
)
( )
(
)
log ln
n
x x
f a bx c d f a bx c d
= + + = + +
The variables a, b, c and d are used here to represent numbers that could possibly be used. To
find the domain of either of these functions, evaluate the following inequality:
0
bx c
+ >
.
Find the domain of each of the following logarithmic functions. Be sure to write the
domain using interval notation.
A)
( )
5
log 2
x
f x
= +
B)
( )
(
)
3
log 4 1 2
x
f x
= − −
C)
( )
2
log 3
x
f x
= −
D)
( )
(
)
2
log 5
x
f x
= −
E)
( )
(
)
5
log 5
x
f x
= − +
F)
( )
ln 4
x
f x
= −
G)
( )
(
)
ln 2 3
x
f x
= −
H)
( )
(
)
3
log 5 1
x
f x
= + +
I)
( )
(
)
ln 3
x
f x
= −
J)
( )
2
log
x
f x
=
K)
( )
3
log
x
f x
=
L)
( )
ln5 1
x
f x
= − +
Explain the difference between the domains of letters J and K above. Although each
problem contains an absolute value, the domains are different. Why is this?
178
Finding the Vertical Asymptote of a Logarithmic Function
( )
(
)
( )
(
)
log ln
n
x x
f a bx c d f a bx c d
= + + = + +
To find the vertical asymptote of a logarithmic function, set bx+c equal to zero and solve. This
will yield the equation of a vertical line. In this case, that vertical line is the vertical
asymptote.
Example
Find the vertical asymptote of the function
( )
(
)
3
log 4 3 2
x
f x
= − −
.
4 3 0
4 3
3
4
x
x
x
− =
=
=
Setting the bx+c term
equal to zero results in
this equation. Simply
solve for x, and we now
have the vertical
asymptote of the function.
Find the vertical asymptote of each of the following logarithmic functions.
A)
( )
5
log 2
x
f x
= +
B)
( )
(
)
3
log 4 1 2
x
f x
= − −
C)
( )
2
log 3
x
f x
= −
D)
( )
(
)
2
log 5
x
f x
= −
E)
( )
(
)
5
log 5
x
f x
= − +
F)
( )
ln 4
x
f x
= −
G)
( )
(
)
ln 2 3
x
f x
= −
H)
( )
(
)
3
log 5 1
x
f x
= + +
I)
( )
(
)
ln 3
x
f x
= −
J)
( )
2
log
x
f x
=
K)
( )
3
log
x
f x
=
L)
( )
ln5 1
x
f x
= − +
179
Graphing Logarithmic Functions
( )
(
)
( )
(
)
log ln
n
x x
f a bx c d f a bx c d
= + + = + +
The parent functions for both logarithmic and natural logarithmic functions look almost
identical. The following functions will be graphed by setting up a table. When faced with the
following function:
( )
3
log
x
f x
=
, find the domain of the function first by evaluating
0
bx c
+ >
. This is done to ensure that only appropriate values are substituted for x. The
values that should be used for x are powers of the base of the logarithm such as
1 1
, , 1, 3, 9.
9 3
and
The values used to substitute for x are completely dependant on the
base of the particular logarithm in the problem. When graphing a natural log with a table,
any number in the domain of the function may be used. Using a calculator, get the decimal
estimates for the y values of the function. DO NOT FORGET to plug in fractional powers for
both logs and natural logs. The fractional powers will give the tail end of the function. Be
sure to plug in the fractions that are right next to the vertical asymptote of the function. Note
in the following picture the graph does not cross the y axis. That is because, in this particular
example,
0
x
=
is the vertical asymptote. By now, you should be aware that the equation
0
x
=
is the equation for the y axis. Remember, always find the vertical asymptote of any
logarithmic function when graphing. This must be done to ensure that it is not crossed. The
range of any normal logarithmic function is all real numbers. The only time this will not be is
the case is that of example K in the finding the domain section on the previous page.
Remember, most people forget the tail end of the
function. Do not forget it. Make sure you note the rise
of the function. The function makes a very slow
gradual rise. It is not steep.
Normally, we will be graphing these functions by means of translations based on the parent
function. This graphing portion is meant to give you practice at using your knowledge about
logarithms and how to use powers to graph the functions. There will be an entire section
devoted to nothing but functions later on in the workbook.
180
Graph each of the following by setting up a table. Identify the range and domain of each
function. Label the x intercept of each function.
We will deal with y intercepts later.
A)
( )
3
log
x
f x
=
x
( )
x
f
B)
( )
4
log 2
x
f x
= −
x
( )
x
f
181
C)
( )
2
log
x
f x
= −
x
( )
x
f
D)
( )
(
)
3
log 1
x
f x
= +
x
( )
x
f
182
E)
( )
ln 3
x
f x
= +
x
( )
x
f
F)
( )
(
)
ln 2
x
f x
= +
x
( )
x
f
183
G)
( )
(
)
3
log
x
f x
= −
x
( )
x
f
H)
( )
2
log
x
f x
=
x
( )
x
f
184
Finding the Inverse of a Function
In the past we worked on “function operations,” in which we added, subtracted, multiplied,
divided, and found composite functions (refer to page 68). The next function operation we
will go over is finding the inverse of a function. If a function is denoted by
( )
x
f
, its’ inverse is
denoted by
( )
1
x
f
−
. The “-1” is a superscript of the term, not an exponent.
Finding the inverse of a function is a 4 step process.
1. Replace
( )
x
f
with y
2. Switch the x and y in the problem.
3. Solve for y
4. Replace y with
( )
1
x
f
−
A function only has an inverse if it is a one-to-one function. This means that for each y value,
there exists only one corresponding x value. Simply put, a function is one-to-one if it passes a
horizontal line test.
Proving whether or not two functions are inverses of each other is a different matter. Two
functions
( )
x
f
and
( )
x
g
are inverses of each other if and only if the following is true:
( )
( )
x
g
f x
=
and
( )
( )
x
f
g x
=
If both composite functions
( )
( )
x
g
f
and
( )
( )
x
f
g
equal x, the functions are inverses of each other.
The official definition states:
( )
( )
x
g
f x
=
for all x in the domain of g.
( )
( )
x
f
g x
=
for all x in the domain of f.
Here is an example of finding the inverse of an exponential function.
( )
3 1
x
x
f
= −
3 1
x
y
= −
Replace
( )
x
f
with y.
3 1
y
x
= −
Switch the x and y.
1 3
y
x
+ =
Solve for y by adding 1 to both sides.
(
)
log 1 log3
y
x + =
Now you need to get to the y exponent, so take the log of both sides.
(
)
log 1 log3
x y+ =
Pull out the exponent y using the properties of logarithms.
(
)
log 1
log3
x
y
+
=
Divide both sides now by
log3
.
(
)
3
log 1
y x
= +
Use the symmetrical property to switch sides.
Using the base change formula, you can rewrite the log with a base of 3.
( )
(
)
1
3
log 1
x
f x
−
= +
Replace y with
( )
1
x
f
−
185
As demonstrated in the previous example, the inverse of an exponential function is a
logarithmic function and vice-versa.
Inverse functions are symmetrical to the
y x
=
axis. When an exponential function and its’
logarithmic inverse are graphed on the same plane, the symmetry is apparent.
Drawing an imaginary diagonal using the line
y x
=
, the functions could literally be folded
along that axis, and the two functions would line up on top of each other.
In the following exercises you will be asked to find the inverse of a logarithmic function. It
should be obvious that the inverse of a logarithmic function is exponential. In order to get to
that point, the problem will need to be rewritten in exponential form.
Logarithmic form Exponential Form
log
a
b c
=
c
a b
=
Here is an example of finding the inverse of a logarithmic function.
( )
(
)
7
log 2 3
x
f x
= − +
(
)
7
log 2 3
y x
= − +
Replace
( )
x
f
with y.
(
)
7
log 2 3
x y
= − +
Switch the x and y.
(
)
7
3 log 2
x y
− = −
Isolate the log by subtracting 3 to both sides.
(
)
7
log 2 3
y x
− = −
Using the symmetrical property, the log is easier to see like this.
3
7 2
x
y
−
= −
Rewrite the log in exponential form.
3
7 2
x
y
−
+ =
Solve for y by adding 2 to both sides.
3
7 2
x
y
−
= +
Again rewrite using the symmetrical property.
( )
1 3
7 2
x
x
f
− −
= +
Replace y with
( )
1
x
f
−
186
Find the inverse of each of the following functions if it exists.
A)
( )
1
3
2
x
f x
= +
B)
( )
4 5
x
f x
= −
C)
( )
2
4
x
x
f
+
=
D)
( )
3
2 1
x
x
f
−
= +
E)
( )
3
3 2
x
f x
= −
F)
( )
( )
2
4 1 3
x
f x
= + −
G)
( )
6
log
x
f x
=
H)
( )
(
)
ln 2
x
f x
= −
I)
( )
4
2log 5
x
f x
= −
J)
( )
4 6
x
f x
= − − +
K)
( )
( )
3
3 2
x
f x
= + −
L)
( )
20
3 1
x
x
f
−
= +
187
Interest Formulas
There are three interest formulas commonly used in mathematics. The first is the standard
interest formula. We will not be using that formula in this section. Instead, we will
concentrate on the other two. We will be working with the compound interest formulas.
There are two formulas for this. The first is a formula for interest being compounded a finite
number of times per year. The second formula is for interest compounded continuously. For
each of these functions, the variable time,
t
, should be in years. Which means 6 months is .5
years.
Standard Interest
Formula
Compound Interest Formula
Interest Compounded
continuously
A= P+Prt
1
nt
r
A P
n
= +
rt
A Pe
=
When using the compound interest formula, what does the variable
n
represent?
When dealing with the compound interest formula, your questions will always come in the
form of word problems. Write down the word associated with the number of times interest
is to be compounded per year when dealing with these word problems.
# of times compounded
Word used in the problem
once
twice
four times
twelve times
twenty-four times
Explain, in your own words, what compounding interest means.
188
If you had a choice to invest a sum of money into an account that yields 7% interest
compounded quarterly verses one that compounds interest semi-annually, which would you
choose?
Answer each of the following.
A)
If you invest $2500 in an account that pays 12% interest, compounded quarterly, how
much would you have at the end of 17 years?
B)
How much would you have to invest in an account that pays 6% interest, compounded
monthly, to have a balance of $30,000 at the end of 10 years?
C)
How long will it take for an investment of $2,000 in an account that pays
1
8
2
% interest
compounded quarterly to become $15,000.
D)
How long will it take for an amount of money to double if deposited in an account that
pays 4.5% interest compounded monthly?
189
E)
At what interest rate must you invest $10,000 to have an ending balance of $72,000 at
the end of 14 years? (Assume interest is compounded quarterly.)
F)
If you invest $26,000 in an account that pays
1
3
2
% interest compounded continuously,
how much would your investment be worth in 18 years.
G)
How long will it take for an amount of money to double if invested in an account that
pays 9% interest compounded continuously.
H)
At what interest rate must you invest $10,000 to have an ending balance of $15,000 at
the end of 9 years? (Assume interest is compounded continuously.)
I)
How much must your principle investment be if you have $30,000 after 16 years in an
account that pays 7.5% interest compounded continuously.
190
J)
Complete the following table to determine the value, A, of an initial investment, P, of
$15,000 into an account that pays 6.5% interest compounded n times for 10 years.
n 1 2 4 12 365
A
K)
Complete the table for the time necessary for an account to triple if interest is
compounded continuously at rate r.
r 2% 3% 4.5% 7% 12%
t
191
Word Problems
A)
The demand equation for a certain clock radio is given by
0.003
400 .06
x
p e= −
. Find the
demand,
x
, for the price of
$99
p
=
.
B)
The population,
P
, where
P
is measured in thousands, of one city is given by
30
kt
P e
=
.
In this particular model,
0
t
=
represents the year 2000. In 1990, the population was
52,000. Find the value of
k
and use the result to estimate the population of the city in
the year 2012.
On the Richter scale, the magnitude
R
of an earthquake with intensity
I
is measured by
10
0
log
I
R
I
=
Where
0
1
I
=
is the minimum intensity used for comparison.
C)
Find the intensity of an earthquake that measures 6.5 on the Richter scale.
D)
Find the intensity of an earthquake the measures 3.2 on the Richter scale.
E)
Find the magnitude of an earthquake that has an intensity of 325,000.
192
Checking Progress
You have now completed the “Logarithms” section of the workbook. The following is a
checklist so that you may check your progress. Check off each of the objectives you have
accomplished.
The student should be able to…
Evaluate a simple logarithm without the aid of a calculator.
Express a logarithmic statement is exponential form.
Express a statement in exponential form in logarithmic form
Expand a logarithmic expression as the sum or difference of logarithms using the
properties of logs.
Condense the sum or difference of logarithms into a single logarithmic expression.
Evaluate logarithms using the base change formula.
Solve logarithmic equations.
Evaluate the solution to logarithmic equations to find extraneous roots.
Solve equations with variables in the exponents.
Find the range and domain of logarithmic functions.
Graph a logarithmic function using a table.
Find the inverse of a function.
Verify two functions are inverses of each other.
Identify a one-to-one function.
Use the compound interest formulas.