Solving Logarithmic Equations Worksheet Page 7
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8Example 5: Solve
log (x 1) log (x 4) 3
+ -
-
=
2
2
This problem contains terms without logarithms.
log (x 1) log (x 4) 3
+ -
-
=
2
2
x 1
+
æ
ö
This problem can be simplified by using Property 4 which
log
=
3
ç
÷
2
changes the subtraction of logarithms to division.
x 4
-
è
ø
x 1
+
Rewrite the problem in exponential form by moving the
3
=
2
base of the logarithm to the other side.
x 4
-
x 1
+
Simplify the problem by cubing the 2.
=
8
x 4
-
Solve for x by crossmultiplying, distributing, subtracting
x 1 8(x 4)
+ =
-
8x from each side, subtracting 1 from each side, and finally
x 1 8x 32
+ =
-
dividing each side by –7.
33
x
=
7
Check the answer; this is an acceptable answer because we
33
x
=
get a positive number when it is plugged back in.
7
33
Therefore, the solution to the problem
log (x 1) log (x 4) 3
+ -
-
= is
x
=
.
2
2
7
log (x 4) log (x 2) log (4x)
+
+
-
=
Example 6: Solve
6
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6
This problem contains only logarithms.
log (x 4) log (x 2) log (4x)
+
+
-
=
6
6
6
This problem can be simplified by using Property 3 which
log ((x 4)(x 2)) log (4x)
+
-
=
6
6
changes the addition of logarithms to multiplication.
Drop the logarithms.
(x 4)(x 2) 4x
+
-
=
Simplify the problem by distributing or FOILing and
2
x
+
2x 8 4x
- =
combining like terms.
Solve the problem by subtracting 4x from each to get it
2
x
-
2x 8 0
- =
equal to zero, and then factoring or using the quadratic
formula to find the values of x.
(x 2)(x 4) 0
+
-
=
x
= -
2 or x
=
4
x
=
4
Check the answers, only one answer is acceptable because
the other answer produces a negative number when it is
plugged back in.
Therefore, the solution to the problem
log (x 4) log (x 2) log (4x)
+
+
-
=
is x = 4.
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