If $A, B$ are real valued (not necessarily independent) random variables with finite means and variances, what do we know about the variance of $\log ( \exp(A) + \exp(B) )$, in terms of the variances of $A, B$? In particular, if the latter are small, will the former be small? Here's my best upper bound so far:
\begin{eqnarray*} \operatorname{VAR}[ \log ( \exp(A) + \exp(B) ) ] &\leq& \operatorname{VAR}[ A ] + \operatorname{VAR}[ B ] + (2 \log 2) \sqrt{\operatorname{VAR}[A] + \operatorname{VAR}[B] } + (\log 2)^2 \end{eqnarray*}
With the bound above, it is not enough for $\operatorname{VAR}[A], \operatorname{VAR}[B]$ to be small for $\operatorname{VAR}\left[ \log\left( \exp(A) + \exp(B) \right) \right]$ to be small because of the $(\log 2)^2$ term. If $A$, $B$ are concentrated each on a single point, then clearly $\operatorname{VAR}[ \log ( \exp(A) + \exp(B) ) ] = 0$. So the bound can be improved, but how? Also, can the bound be improved if we know that $A$ and $B$ are in fact dependent?
Here's how to derive the current bound:
\begin{eqnarray} \operatorname{VAR}[ \log ( \exp(A) + \exp(B) ) ] &=& \operatorname{VAR}[ \log( \exp( \max\{A,B\} ) + \exp(\min\{A,B\} )) ] \\ &=& \operatorname{VAR}[ \log( \exp( \max\{A,B\} ) ( 1 + \exp(\min\{A,B\} - \max\{A,B\})]\\ &=& \operatorname{VAR}[ \max\{A,B\} ] \\ & & + \operatorname{VAR}[ \log( 1 + \exp(\min\{A,B\} - \max\{A,B\}))] \\ & & + 2 \operatorname{COV}[ \max\{A,B\} , \log( 1 + \exp(\min\{A,B\} - \max\{A,B\})) ] \end{eqnarray} For the first term, we will use \begin{eqnarray} \operatorname{VAR}[ \max\{A,B\} ] \leq \operatorname{VAR}[ A ] + \operatorname{VAR}[ B ] \end{eqnarray} (see https://stats.stackexchange.com/q/48093 ). Define \begin{eqnarray} C \stackrel{\Delta}{=} \log(1+ \exp(\min\{A,B\} - \max\{A,B\}) ) \end{eqnarray} and note that $0 \leq C \leq \log 2$. The second term is bounded this way: \begin{eqnarray} \operatorname{VAR}[ C ] &=& E C^2 - (E C)^2 \leq E C^2 \leq (\log 2)^2 \end{eqnarray} Finally, for the third term \begin{eqnarray*} \operatorname{COV}[ \max\{A,B\} , C ] &\leq & \sqrt{ \operatorname{VAR}[\max\{A,B\}] \operatorname{VAR}[C]} \\ &\leq & \sqrt{ (\operatorname{VAR}[A] + \operatorname{VAR}[B] ) (\log 2)^2} \\ &=&(\log 2) \sqrt{\operatorname{VAR}[A] + \operatorname{VAR}[B] } \end{eqnarray*}
Putting everything together, this currently reads:
\begin{eqnarray*} \operatorname{VAR}[ \log ( \exp(A) + \exp(B) ) ] &\leq& \operatorname{VAR}[ A ] + \operatorname{VAR}[ B ] + (2 \log 2) \sqrt{\operatorname{VAR}[A] + \operatorname{VAR}[B] } + (\log 2)^2 \end{eqnarray*}
P.S. The idea works more generally: \begin{eqnarray*} \operatorname{VAR}\left[ \log\left( \sum_{i=1}^n \exp(X_i) \right) \right] \leq \sum_{i=1}^n \operatorname{VAR}[ X_i ] + (2 \log n) \sqrt{\sum_{i=1}^n \operatorname{VAR}[X_i]} + (\log n)^2 \end{eqnarray*} by observing that if $Y_1, \cdots, Y_n$ are ordered versions of $X_1,\cdots,X_n$, (largest to smallest) \begin{eqnarray*} \sum_{i=1}^n \exp(X_i) &=& \sum_{i=1}^n \exp(Y_i) \\ &=& \exp(Y_1) \left( 1 + \sum_{i=2}^n \exp(Y_i - Y_1) \right) \end{eqnarray*}