Solving Logarithmic Equations Worksheet Page 6
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8Example 3: Solve
log (2x 1) log (x 2) log 3
+
=
+
-
4
4
4
This problem contains only logarithms.
log (2x 1) log (x 2) log 3
+
=
+
-
4
4
4
x 2
+
This problem can be simplified by using Property 4 which
æ
ö
log (2x 1) log
+
=
ç
÷
4
4
changes the subtraction of logarithms to division.
3
è
ø
x 2
+
Drop the logarithms.
2x 1
+ =
3
Simplify the problem by crossmultiplying to get rid of the
3(2x 1)
+
= +
x 2
fractions.
Solve the problem by distributing the 3, subtracting x from
6x 3 x 2
+ = +
each side, subtracting 3 from each side, and finally dividing
1
by 5.
x
= -
5
No Solution
Check the answers, this problem has “No Solution”
because the only answer produces a negative number and
we can’t take the logarithm of a negative number.
Therefore, the problem
log (2x 1) log (x 2) log 3
+
=
+
-
has no solution.
4
4
4
Example 4: Solve log(5x 11) 2
-
=
This problem contains terms without logarithms.
log(5x 11) 2
-
=
This problem does not need to be simplified because there
log(5x 11) 2
-
=
is only one logarithm in the problem.
Rewrite the problem in exponential form by moving the
2
5x 2 10
- =
base of the logarithm to the other side. For common
logarithms the base is 10.
Simplify the problem by squaring the 10.
5x 2 100
- =
Solve for x by adding 2 to each side and then dividing each
102
x
=
side by 5.
5
Check the answer; this is an acceptable answer because we
102
x
=
get a positive number when it is plugged back in.
5
102
Therefore, the solution to the problem log(5x 11) 2
-
= is
x
=
.
5
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