Logarithmic Equations
A
logarithmic equation
is an equation containing a variable in a logarithmic
expression. Examples of logarithmic equations include
Some logarithmic equations can be expressed in the form
We can
solve such equations by rewriting them in exponential form.
log
b
M
=
c
.
log
4
1
x
+
3
2
=
2
and
ln
1
x
+
2
2

ln
1
4
x
+
3
2
=
ln
a
1
x
b
.
e
2
x

8
e
x
+
7
=
0.
5
0, ln 3
6
.
ln
e
x
=
x
x
=
ln 3
ln
e
x
=
ln
3
x
=
0
e
x
.
e
x
=
3
e
x
=
1
e
x

3
=
0
or
e
x

1
=
0
u
=
e
x
,
u
2

4
u
+
3
=
1
u

3
21
u

1
2
.
1
e
x

3
21
e
x

1
2
=
0
e
2
x

4
e
x
+
3
=
0
u
2

4
u
+
3
=
0.
u
=
e
x
,
e
2
x

4
e
x
+
3
=
0.
EXAMPLE 5
Use the definition of a logarithm
to solve logarithmic equations.
Using the Definition of a Logarithm to Solve Logarithmic Equations
1.
Express the equation in the form
2.
Use the definition of a logarithm to rewrite the equation in exponential form:
3.
Solve for the variable.
4.
Check proposed solutions in the original equation. Include in the solution set
only values for which
M
7
0.
log
b
M=c
means
b
c
=M.
Logarithms are exponents.
log
b
M
=
c
.
Solving Logarithmic Equations
Solve:
a.
b.
Solution
The form
involves a single logarithm whose coefficient is
1 on one side and a constant on the other side. Equation (a) is already in this form.
We will need to divide both sides of equation (b) by 3 to obtain this form.
log
b
M
=
c
3 ln
1
2
x
2
=
12.
log
4
1
x
+
3
2
=
2
EXAMPLE 6
a.
This is the given equation.
Rewrite in exponential form:
means
Square 4.
Subtract 3 from both sides.
Check
13
:
This is the given logarithmic equation.
Substitute 13 for
true
because
This true statement indicates that the solution set is
b.
This is the given equation.
Divide both sides by 3.
Rewrite the natural logarithm showing base
This step is
optional.
Rewrite in exponential form:
means
Divide both sides by 2.
Check
This is the given logarithmic equation.
Substitute
for
Simplify:
Because ln
we conclude ln
true
This true statement indicates that the solution set is
Check Point
6
Solve:
a.
b.
Logarithmic expressions are defined only for logarithms of positive real
numbers.
Always check proposed solutions of a logarithmic equation in the original
equation. Exclude from the solution set any proposed solution that produces the
logarithm of a negative number or the logarithm of 0.
To rewrite the logarithmic equation
in the equivalent exponential
form
we need a single logarithm whose coefficient is one. It is sometimes
necessary to use properties of logarithms to condense logarithms into a single
logarithm. In the next example, we use the product rule for logarithms to obtain a
single logarithmic expression on the left side.
Solving a Logarithmic Equation
Solve:
Solution
This is the given equation.
Use the product rule to obtain a single
logarithm: log
b
M
+
log
b
N
=
log
b
1
MN
2
.
log
2
3
x
1
x

7
24
=
3
log
2
x
+
log
2
1
x

7
2
=
3
log
2
x
+
log
2
1
x

7
2
=
3.
EXAMPLE 7
b
c
=
M
,
log
b
M
=
c
4 ln
1
3
x
2
=
8.
log
2
1
x

4
2
=
3
b
e
4
2
r
.
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