To solve the equation [tex]\(9^{x+4} = 11\)[/tex] for [tex]\(x\)[/tex] using logarithms, we follow these steps:
1. Express the equation in logarithmic form:
[tex]\[
9^{x+4} = 11
\][/tex]
To solve for [tex]\(x\)[/tex], we take the logarithm of both sides of the equation. It is common to use base 10 logarithms (denoted as [tex]\(\log\)[/tex]):
[tex]\[
\log(9^{x+4}) = \log(11)
\][/tex]
2. Apply the power rule of logarithms:
The power rule states that [tex]\(\log(a^b) = b \cdot \log(a)\)[/tex]. Applying this to our equation, we have:
[tex]\[
(x+4) \cdot \log(9) = \log(11)
\][/tex]
3. Isolate the term involving [tex]\(x\)[/tex]:
To isolate [tex]\(x\)[/tex], we divide both sides by [tex]\(\log(9)\)[/tex]:
[tex]\[
x + 4 = \frac{\log(11)}{\log(9)}
\][/tex]
4. Calculate the values of the logarithms:
From the given solution, we know:
[tex]\[
\log(11) \approx 2.3979
\][/tex]
and
[tex]\[
\log(9) \approx 2.1972
\][/tex]
5. Substitute the logarithm values in the equation:
[tex]\[
x + 4 = \frac{2.3979}{2.1972}
\][/tex]
6. Simplify the fraction:
[tex]\[
x + 4 \approx 1.0913
\][/tex]
7. Solve for [tex]\(x\)[/tex]:
Finally, subtract 4 from both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 1.0913 - 4
\][/tex]
[tex]\[
x \approx -2.909
\][/tex]
Thus, the solution to the equation [tex]\(9^{x+4} = 11\)[/tex] is [tex]\(x \approx -2.909\)[/tex].
