2log 2 81 log 4 2 log8 4 log16raise to 8 is
(A) 2
(B)3
(C) 4
(D) None of these
Answers
Answer:
WARNING: This is not a logarithmic equation. In fact, the arguments of the logarithms are constants (numbers), not variables. So this is a simple linear equation with real (logarithmic) coefficients.
We can treat the coefficients as with integer or rational coefficients. We try to rearrange the equation to have all the terms that depend on
x
on one side and "the numbers" on the other side:
x
log
4
+
x
log
2
=
log
10
−
log
8
x
(
log
4
+
log
2
)
=
log
10
−
log
8
x
=
log
10
−
log
8
log
4
+
log
2
Now we can use the following two properties of logarithms to simplify the expression:
log
a
+
log
b
=
log
(
a
⋅
b
)
log
c
−
log
d
=
log
(
c
d
)
In our case
a
=
4
,
b
=
2
,
c
=
10
and
d
=
8
, so
log
4
+
log
2
=
log
6
and
log
10
−
log
8
=
log
(
10
8
)
=
log
(
5
4
)
We end with
x
=
log
6
log
(
5
4
)
Note
We can apply the change-of-base formula too:
log
p
log
q
=
log
q
p
The result will be written in an even more compact way, but the base of the logarithm would be quite unpractical. In fact, in our case
p
=
6
and
q
=
5
4
. We get
x
=
log
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