Answer:
C. -34.972
Explanation:
Given the equation:
[tex]9^{x+4}=7^x[/tex]
To solve for x, follow the steps below.
Step 1: Take the logarithm of both sides.
[tex]\log (9^{x+4})=\log (7^x)[/tex]
Step 2: Apply the index of a logarithm law stated below.
[tex]\begin{gathered} \log (a^n)=n\log (a) \\ \implies\log (9^{x+4})=\log (7^x) \\ (x+4)\log 9=x\log (7) \end{gathered}[/tex]
Step 3: Bring all the terms containing x together by dividing both sides by x log (9)
[tex]\begin{gathered} \frac{(x+4)\log(9)}{x\log(9)}=\frac{x\log(7)}{x\log(9)} \\ \implies\frac{x+4}{x}=\frac{\log(7)}{\log(9)} \end{gathered}[/tex]
Step 4: Use a calculator to evaluate the right-hand side.
[tex]\frac{x+4}{x}=0.885622[/tex]
Step 5: Cross multiply and solve for x.
[tex]\begin{gathered} x+4=0.885622x \\ x-0.885622x=-4 \\ 0.114378x=-4 \\ \frac{0.114378x}{0.114378}=\frac{-4}{0.114378} \\ x=-34.972 \end{gathered}[/tex]
The value of x is -34.972 (Option C).
