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im terribly confused as to how i would apply common logarithms to this problem. :(

|dw:1386036292570:dw| i ended up with this

|dw:1386036395791:dw| would it turn to this?

I ended up with \[4^x= \frac{ 8+\sqrt{52} }{ 2 }\]

The answer i should end up with\[\frac{ \log(4+\sqrt{19)} }{\log4 }\]

Check your math in the quadratic formula, looks like your numbers aren't right.

\[4^x=4+\sqrt{19}\]
\[x \log_{10}4=\log_{10}(4+\sqrt{19}) \]

Divide both sides by log 4

I am so grateful to both of you. Thank you so much! :)