Sure, let’s solve the equation [tex]\( 3^{x+2} = 15 \)[/tex] for [tex]\( x \)[/tex] using the change of base formula for logarithms.
1. Understand the equation:
We have [tex]\( 3^{x+2} = 15 \)[/tex]. Our goal is to isolate [tex]\( x \)[/tex].
2. Take the logarithm of both sides:
To simplify, we can take the natural logarithm (or any logarithm base) of both sides of the equation:
[tex]\[
\log(3^{x+2}) = \log(15)
\][/tex]
3. Apply the power rule of logarithms:
The logarithm power rule states that [tex]\( \log(a^b) = b \log(a) \)[/tex]. Applying this to the left side:
[tex]\[
(x+2) \log(3) = \log(15)
\][/tex]
4. Solve for [tex]\( x + 2 \)[/tex]:
To isolate [tex]\( x + 2 \)[/tex], divide both sides by [tex]\( \log(3) \)[/tex]:
[tex]\[
x+2 = \frac{\log(15)}{\log(3)}
\][/tex]
5. Calculate the value:
Using the change of base formula [tex]\( \log_b(y) = \frac{\log(y)}{\log(b)} \)[/tex]:
[tex]\[
x+2 = \frac{\log(15)}{\log(3)}
\][/tex]
The calculation of [tex]\( \frac{\log(15)}{\log(3)} \)[/tex] results in approximately [tex]\( 2.464973520717927 \)[/tex].
6. Isolate [tex]\( x \)[/tex]:
Now, subtract 2 from both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{\log(15)}{\log(3)} - 2
\][/tex]
Which yields:
[tex]\[
x \approx 2.464973520717927 - 2 \approx 0.4649735207179271
\][/tex]
So, the solution to the equation [tex]\( 3^{x+2} = 15 \)[/tex] for [tex]\( x \)[/tex] is approximately [tex]\( 0.465 \)[/tex].
Therefore, the closest answer choice is:
[tex]\[ \boxed{0.465} \][/tex]
