What did I do wrong with this logarithmic equation?

The calculation (last step) step is wrong.

$$ x = \frac{\ln(4e) + 2}{2} = \frac{\ln 2^{2} + \ln e + 2}{2} = \frac{2\ln 2 + 3}{2} = \ln 2 + \frac{3}{2} \approx 2.193 $$

(Also, you should have used parentheses the third step down.)

As indicated by the third line of your computation, you were careless with your parentheses. What you computed is $$ \frac{\ln(4)\cdot\mathrm{e} + 2}{2} \approx 2.884. $$ My guess is that what you typed into the calculator was something like

 ( ln 4 * e + 2 ) / 2

or possibly

 ln 4 * e + 2 / 2

depending on what calculator or computer program you are using.

Your first edit also does not fix the issue. Your calculator was not assuming that you meant $$ \frac{\ln(4\mathrm{e} + 2)}{2} \approx 2.996. $$ What you meant to calculate was $$ \frac{\ln(4\mathrm{e}) + 2}{2} \approx 2.193. $$ On most calculators, this would be something like

 ( ln ( 4 * e ) + 2 ) / 2

Parentheses matter. Be careful with them.