To solve the equation [tex]\(6^{x+2} = 10\)[/tex] for [tex]\(x\)[/tex] using the change of base formula [tex]\(\log_b y = \frac{\log y}{\log b}\)[/tex], let's follow these steps:
1. Express the equation in logarithmic form:
[tex]\[
6^{x+2} = 10
\][/tex]
2. Take the natural logarithm (or any logarithm) on both sides:
[tex]\[
\log(6^{x+2}) = \log(10)
\][/tex]
3. Use the logarithm power rule [tex]\(\log(a^b) = b \log(a)\)[/tex]:
[tex]\[
(x+2) \log(6) = \log(10)
\][/tex]
4. Solve for [tex]\(x+2\)[/tex]:
[tex]\[
x+2 = \frac{\log(10)}{\log(6)}
\][/tex]
5. Isolate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{\log(10)}{\log(6)} - 2
\][/tex]
6. Calculate the numerical value:
[tex]\[
x = -0.715
\][/tex]
Therefore, the solution to the equation [tex]\(6^{x+2} = 10\)[/tex] is [tex]\(x = -0.715\)[/tex].
The correct multiple-choice answer is: [tex]\(-0.715\)[/tex]
