Solving Logarithmic Equations Worksheet Page 5
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8Examples – Now let’s use the steps shown above to work through some examples. These examples will be a
mixture of logarithmic equations containing only logarithms and logarithmic equations containing terms
without logarithms.
Example 1: Solve
log (9x 2) 4
+
=
3
This problem contains terms without logarithms.
log (9x 2) 4
+
=
3
This problem does not need to be simplified because there
log (9x 2) 4
+
=
3
is only one logarithm in the problem.
Rewrite the problem in exponential form by moving the
4
9x 2 3
+ =
base of the logarithm to the other side.
Simplify the problem by raising 3 to the fourth power.
9x 2 81
+ =
Solve for x by subtracting 2 from each side and then
79
x
=
dividing each side by 9.
9
Check the answer; this is an acceptable answer because we
79
x
=
get a positive number when it is plugged back in.
9
79
Therefore, the solution to the problem
log (9x 2) 4
+
= is
x
=
.
3
9
Example 2: Solve
log x log (x 12) 3
+
-
=
4
4
This problem contains terms without logarithms.
log x log (x 12) 3
+
-
=
4
4
This problem can be simplified by using Property 3 which
log (x(x 12)) 3
-
=
4
changes the addition of logarithms to multiplication.
Rewrite the problem in exponential form by moving the
3
x(x 12)
-
=
4
base of the logarithm to the other side.
Simplify the problem by distributing and cubing the 4.
2
x
-
12x 64
=
Solve the problem by subtracting 64 from each to get it
2
x
-
12x 64 0
-
=
equal to zero, and then factoring or using the quadratic
(x 4)(x 16) 0
+
-
=
formula to find the values of x.
x
= -
4 or x 16
=
Check the answers, only one answer is acceptable because
x 16
=
the other answer produces a negative number when it is
plugged back in.
Therefore, the solution to the problem
log x log (x 12) 3
+
-
= is x = 16.
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