It's hard to read the attachment, but it looks like the first line is
$\displaystyle h^2 + k^2 = 23 hk$
Please correct me if I misread that. Then on the second line you have written:
$\displaystyle \log(\frac {h+k} 5 ) = \frac 1 2 ( \log h + \log k) $
Not sure how you got that, but in general it's not true. Using the identity: $\displaystyle \log( \frac a b ) \ \log a - \log b$ in general:
$\displaystyle \log(\frac {h+k} 5 ) = \log(h+k) - \log 5$
It would help if can tell us what it is you are trying do with this problem.
you worked the problem backwards. You need to start with $\displaystyle h^2 + k^2 = 23 h k$ as in
$\displaystyle h^2 + k^2 = 23 h k \Rightarrow$
$\displaystyle (h+k)^2=25 h k \Rightarrow$
$\displaystyle \left(\frac{h+k}{5}\right)^2=hk \Rightarrow$
$\displaystyle 2\log\left(\frac{h+k}{5}\right)=\log(h)+\log(k) \Rightarrow$
$\displaystyle \log\left(\frac{h+k}{5}\right)=\frac{\log(h)+\log( k)}{2}$
You didn't really do it backwards - that would have meant starting with $\displaystyle \log ( \frac {h+k}5 ) = \frac 1 2 (( \log h + \log k)$ and working from that to arrive at $\displaystyle h^2 + k^2 = 23 hk$. Your approach was a bit different - you started working backwards but mid-way threw in the initial given equation to ultimately arrive at 1 = 1, and hence conclude that the original statement must therefore be true. That works, but to avoid confusion like I had when I first saw your work you should notate your work to help the reader folllow your logic. The problem is that if you simply write a series of equations the reader assumes that each equation comes naturally from the one before it. But in your work it doesn't flow like that, so it's confusing. It's best to start with the given fact that $\displaystyle h^2 + k^2 = 23hk$ and from that proceed through a series of steps where each derives from the previous, to ultimately arrive at the desired expression.