Answer
$x=\frac{8\ln(10)-\ln(3)}{\ln 2}$
Work Step by Step
Exponentiate both sides with base $10$ to get $$10^{\log(3 \cdot 2^x)}=10^8$$ $$3 \cdot 2^x = 10^8$$ Next, divide both sides by $3$ to get $$2^x=\frac{10^8}{3}$$ Take the natural logarithm of both sides to get $$\ln(2^x)=\ln(10^8/3)$$ Since $\ln a^b=b \ln a$ and $\ln(a/b)=\ln(a)-\ln(b)$, the equation becomes $$x \ln 2 = 8 \ln(10)-\ln(3)$$ Divide both sides by $\ln 2$ to get $$x=\frac{8 \ln(10) - \ln(3)}{\ln 2}$$
