Equations of this form are solved by first taking the natural logarithm (ln) of both sides. So:
[tex]\ln (3^{x+6})=\ln (8)[/tex]We now use the property of logarithms shown below:
[tex]\ln (a^b)=b\ln (a)[/tex]So, the equation becomes:
[tex]\begin{gathered} \ln (3^{x+6})=\ln (8) \\ (x+6)\ln (3)=\ln (8) \end{gathered}[/tex]Distributing the value ln(3), we get:
[tex]\begin{gathered} (x+6)\ln (3)=\ln (8) \\ \ln (3)x+6\ln (3)=\ln (8) \end{gathered}[/tex]Now, we solve for x:
[tex]\begin{gathered} \ln (3)x+6\ln (3)=\ln (8) \\ \ln (3)x=\ln (8)-6\ln (3) \\ x=\frac{\ln (8)-6\ln (3)}{\ln (3)} \\ x=\frac{\ln 8}{\ln 3}-6 \end{gathered}[/tex]Looking at the answer choices,
D is the correct answer!