I need help in solving logarithms question...?

Question

My question is:
(log7 27)(log3 49)

where 7 & 3 are subscript)

Answer 1

(log7 27)(log3 49)

The answer is 6, and it can be done without a calculator. Doesn't it look kind of suspicious that you are taking log-base-3 of a power of 7 (49 = 7^2), and log-base-7 of a power of 3 (27 = 3^3)?

The thing you need to know to get the answer is that logx(y) = log(y)/log(x). You can use plain base-10 logs, or natural log, either way it works. Dividing by the log of the base is how you change bases in logs.

So... your expression can be rewritten as:

(log7 27)(log3 49) =
log(27)/log(7) * log(49)/log(3)

But we can multiply that out and refactor it, switching the denominators:

log(27)/log(7) * log(49)/log(3) =
log(27)log(49) / log(7)log(3) =
log(27)/log(3) * log(49)/log(7)

And if we convert back from log(y)/log(x) to logx(y), we now have log-base-3 of a power of 3 and log-base-7 of a power of 7, which can be done easily (logx(x^y) = y). It simplifies as:

log(27)/log(3) * log(49)/log(7) =
log3(27) * log7(49) =
log3(3^3) * log7(7^2) =
3*log3(3) * 2*log7(7) =
3*1 * 2*1 =
6

Answer 2

This question looks as if it should read:-
(log3 27) (log7 49)
= 3 x 2
= 6

Answer 3

When you have a log that is not base 10 (your normal log), They you can re-write it this way:

loga b = (log10 b) / (log10 a) = (log b) / (log a)

So, in the case of your problem, you can re-write yours to look like:

log7 27 = (log10 27) / (log10 7) = (log 27) / (log 7)
log3 49 = (log10 49) / (log10 3) = (log 49) / (log 3)

So, plug these in and use your calculator...

(log7 27)(log3 49)
= [(log 27) / (log 7)] * [(log 49) / (log 3)]
= (1.431 / 0.845 ) * (1.690 / 0.477)
= 3.967 / 4.444
= 0.893

Answer 4

=(log27/log7)(log49/log3)=3x2 = 6