math homework: logarithms?

Question

3/2 (log7)X=(log7)125

the sevens are subscript. I couldn't figure out how to do on the computer. help me please!! show work!!

Thanks a lot!! XD

Answer 1

a * log b = log (b^a)

So...

3/2 (log7)X=(log7)125
... implies that:
log7(x^(3/2)) = log7(125)

If you are told that log(a) = log(b), then a=b, so:

log7(x^(3/2)) = log7(125)
... implies that:
x^(3/2) = 125

Now you've gotten rid of the logs, and you need to find x such that x^(3/2) = 125.

x^(3/2) is the same as (sqrt(x))^3, so...

x^(3/2) = 125
... implies that:
(sqrt(x))^3 = 125

... 5^3 = 125, so, taking the cube root of both sides:

(sqrt(x))^3 = 125
sqrt(x) = 5

If sqrt(x) = 5, then x=25.

Answer 2

In the following, all logs are to base 7:-
(3/2).log X = log (125)
log X = (2/3).log (125)
log X = log [125^(2/3) ]
X = 125^(2/3)
X = 5²
X = 25

Answer 3

(3/2) log[base 7](x) = log[base 7](125)

First, bring the (3/2) into the log as a power.

log[base 7](x^(3/2)) = log[base 7](125)

Take the antilog of both sides.

x^(3/2) = 125

Bring both sides to the power of (2/3), to isolate x.

x = 125^(2/3)

Which is the same as

x = (125^(1/3))^2
x = 5^2
x = 25