I distributed the natural logarithm and got $(0 + 0.549)$ [placing the values in a calculator]. However, the answer key states that the answer is $1.0051$.
Where did I go wrong?
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You are confusing your identities. You are most likely confusing the identity $\ln(ab)=\ln a+\ln b$. In general $\ln(a+b)\neq\ln a+\ln b$. Also, note that if you are using a calculator, you do not need to simplify the expression; you can punch $\ln(1+\sqrt 3)$ straight into your calculator (but don't forget the brackets!).
As someone previously stated, logarithms are not linear. It can be confusing to keep the properties straight, but a good logic check is that if:
$$ \log(a+b) = \log(a)+\log(b)\\ \rightarrow \log(a)=\log(0+a)=\log(0)+\log(a)\space\space(a\neq0)\\\rightarrow \log(a)~ DNE ~~ \forall a $$ Which is obviously incorrect.
Hint: Use the Taylor series of $\ln(1+x)$ at the point $x=1$. See related problem.