How can the logarithm problem be solved?

[chwala]

Homework Statement

Given that, $$log_p X=5$$ and $$log_p Y=2$$.
Find $$log_{xy} P$$

Relevant Equations

Logs

Interesting, i have not worked on logs with conjoined bases before, anyway my approach is as follows;

$$p^5=x$$ and $$p^2=y$$
Let $$log_{xy}P = m$$, →$$(xy)^m = P$$
$$(P^5⋅P^2)^m = P^1$$
$$P^{7m}=P^1$$
$$m=\frac {1}{7}$$

Any other way of looking at this is most welcome.

 

[anuttarasammyak]

As you find
\log_ab \ \log_ba=1

 

[pasmith]

\log_{xy}(p) = \frac{\log_p(p)}{\log_p(xy)} = \frac{1}{2 + 5}.

 

[mjc123]

You are, of course, assuming that x = X, y = Y and p = P. Or just being inconsistent in your notation.

 

[chwala]

I have been thinking about this question and my bad, i did not indicate the letters in the right manner. Find the question below as it appears on the textbook;

This implies that, my method was after all correct! Sorry for my lack of details here...

 

[chwala]

You are, of course, assuming that x = X, y = Y and p = P. Or just being inconsistent in your notation.

Kindly check my post $5$...i did not capture the question correctly...My apologies mjc...