How do I simplify $\frac{\log_7 32}{\log_7 8\cdot\sqrt2}$? So far I have got $\log_7 2^5 - \log_7 2^3 + \log_7 2^{1/2}$ and am unable to proceed. Am I, on the right track so far? and how do I proceed? thank you!

You can use the formula
$$\frac{\log_{a}b}{\log_{a}c}=\log_{c}b$$
$$\frac{\log_{7}32}{\log_{7}8}\cdot \sqrt{2}=\left(\log_{8}32\right)\cdot \sqrt{2}=\frac{5\sqrt{2}}{3}$$
Or If you are looking for this 
$$\frac{\log_{7}32}{\log_{7}8\sqrt{2}} =\left(\log_{8\sqrt{2}}32\right)=\frac{5}{\frac{7}{2}}=\frac{10}{7}$$
