Question

Solve the exponential equation. Express the solution of natural logarithmic

Answer 1

`3^(x+6)=8`

Equations of this form are solved by first taking the natural logarithm (ln) of both sides. So:

`\ln (3^(x+6))=\ln (8)`

We now use the property of logarithms shown below:

`\ln (a^b)=b\ln (a)`

So, the equation becomes:

`\begin{gathered} \ln (3^(x+6))=\ln (8) \\ (x+6)\ln (3)=\ln (8) \end{gathered}`

Distributing the value ln(3), we get:

`\begin{gathered} (x+6)\ln (3)=\ln (8) \\ \ln (3)x+6\ln (3)=\ln (8) \end{gathered}`

Now, we solve for x:

`\begin{gathered} \ln (3)x+6\ln (3)=\ln (8) \\ \ln (3)x=\ln (8)-6\ln (3) \\ x=(\ln (8)-6\ln (3))/(\ln (3)) \\ x=(\ln 8)/(\ln 3)-6 \end{gathered}`

Looking at the answer choices,

D is the correct answer!