Vtyjkyuk

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!

Your expansion is not correct. If $b, x, y > 0$, with $b neq 1$, then
beginalign*
log_b (xy) & = log_b x + log_b y\
log_b left(fracxyright) & = log_b x - log_b y
endalign*
However, you have
$$fraclog_7 32log_7 8sqrt2$$
which is a quotient of logarithms, not the logarithm of a quotient, so you cannot apply those rules here.

However, your observations that $32 = 2^5$, $8 = 2^3$, and $sqrt2 = 2^1/2$ would be useful if we were working with logarithms to the base $2$. To do so, we use the Change of Base Formula. Let $a, b, x > 0$, with $a, b neq 1$. Then
$$log_a x = fraclog_b xlog_b a$$
By setting $a = 7$ and $b = 2$, we obtain
beginalign*
fraclog_7 32log_7 8sqrt2 & = fracdfraclog_2 32log_2 7dfraclog_2 8sqrt2log_2 7\
& = fraclog_2 32log_2 8sqrt2\
& = fraclog_2 2^5log_2 2^32^1/2\
& = frac5log_2 2^7/2\
& = frac5frac72\
& = frac107
endalign*

You can use the formula

$$fraclog_ablog_ac=log_cb$$
$$fraclog_732log_78cdot sqrt2=left(log_832right)cdot sqrt2=frac5sqrt23$$

Or If you are looking for this

$$fraclog_732log_78sqrt2 =left(log_8sqrt232right)=frac5frac72=frac107$$