Answer
$$
\frac{d}{d t} \log _{10}( t+2^t)=\frac{1+2^t \ln 2}{ (\ln 10)( t+2^t) }.
$$
Work Step by Step
Recall that $(\log_b x)'=\dfrac{1}{(\ln b)x}$
Recall that $(a^x)'=a^x\ln{a}$
Thus we have:
$$
\frac{d}{d t} \log _{10}( t+2^t)=\frac{1+2^t \ln 2}{ (\ln 10)( t+2^t) }.
$$
