To solve the equation [tex]\(4^{x+2}=12\)[/tex] for [tex]\(x\)[/tex] using the change of base formula [tex]\(\log_b y = \frac{\log y}{\log b}\)[/tex], follow these steps:
1. Apply the natural logarithm to both sides of the equation:
[tex]\[
\ln(4^{(x+2)}) = \ln(12)
\][/tex]
2. Use the power rule for logarithms, which states [tex]\(\ln(a^b) = b \ln(a)\)[/tex]:
[tex]\[
(x+2) \ln(4) = \ln(12)
\][/tex]
3. Solve for [tex]\(x + 2\)[/tex]:
[tex]\[
x + 2 = \frac{\ln(12)}{\ln(4)}
\][/tex]
4. Calculate the numerical value:
[tex]\[
x + 2 \approx \frac{2.4849066497880004}{1.3862943611198906} \approx 1.7924812503605783
\][/tex]
5. Isolate [tex]\(x\)[/tex] by subtracting 2 from both sides:
[tex]\[
x = 1.7924812503605783 - 2 \approx -0.20751874963942174
\][/tex]
Hence, the solution to the equation [tex]\(4^{x+2} = 12\)[/tex] for [tex]\(x\)[/tex] is approximately [tex]\(-0.207519\)[/tex].
Thus, the correct multiple-choice answer is:
[tex]\[
-0.207519
\][/tex]
