To solve the equation [tex]\(9^{x+4} = 11\)[/tex], we will use logarithms to simplify and solve for [tex]\(x\)[/tex]. Follow these steps:
1. Apply the logarithm to both sides of the equation:
[tex]\[
\log(9^{x+4}) = \log(11)
\][/tex]
2. Use the power rule of logarithms which states [tex]\(\log(a^b) = b \log(a)\)[/tex]:
[tex]\[
(x+4) \log(9) = \log(11)
\][/tex]
3. Isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
x + 4 = \frac{\log(11)}{\log(9)}
\][/tex]
4. Calculate the right-hand side:
[tex]\(\log(11)\)[/tex] and [tex]\(\log(9)\)[/tex] are standard logarithms which can be calculated using a calculator; the precise values of [tex]\(\log(11)\)[/tex] and [tex]\(\log(9)\)[/tex] are:
[tex]\[
\log(11) \approx 1.041393
\][/tex]
[tex]\[
\log(9) \approx 0.954243
\][/tex]
So,
[tex]\[
\frac{\log(11)}{\log(9)} \approx \frac{1.041393}{0.954243} \approx 1.091329
\][/tex]
5. To isolate [tex]\(x\)[/tex], subtract 4 from both sides:
[tex]\[
x = 1.091329 - 4
\][/tex]
[tex]\[
x \approx -2.908671
\][/tex]
Now, we need to compare this solution with the given potential answers to determine the closest one. The possible answers are:
- [tex]\(-3.094\)[/tex]
- [tex]\(-2.909\)[/tex]
- [tex]\(4.916\)[/tex]
- [tex]\(5.091\)[/tex]
The solution we found is approximately [tex]\(-2.908671\)[/tex]. Among the choices, the closest to [tex]\(-2.908671\)[/tex] is [tex]\(-2.909\)[/tex].
Therefore, the nearest solution to [tex]\(x\)[/tex] is:
[tex]\[
\boxed{-2.909}
\][/tex]
