Before I start I would like to suggest that you repeat the problem statement in the written portion of your post. It would be easier for us to see what you are trying to write, particularly when we get LaTeX back.

I presume that "log a b^2" is "log to the base a of b^2? Let me rewrite that as log(a)b^2.

[log(a)b^2]*[log(b)a^3] = [2*log(a)b]*[3*log(b)a] = 6*[log(a)b]*[log(b)a]

There are probably several ways to finish this. I am going to use the change of base formula to change both of these to log base 10 (which I will write simply as "log x.")

The change of base formula is:

log(k)x = (log x)/(log k) to convert to log base 10.

So

log(a)b = (log b)/(log a)

log(b)a = (log a)/(log b)

So

[log(a)b^2]*[log(b)a^3] = 6*[log(a)b]*[log(b)a] = 6*(log b)/(log a)*(log a)/(log b) = 6

-Dan