To solve the equation [tex]\(3^{x+1} = 15\)[/tex] for [tex]\(x\)[/tex], we can use logarithms and the change of base formula [tex]\(\log_b(y) = \frac{\log(y)}{\log(b)}\)[/tex]. Here is a detailed step-by-step solution:
1. Step 1: Apply the natural logarithm to both sides.
Take the natural logarithm (logarithm with base [tex]\(e\)[/tex]) of both sides of the given equation:
[tex]\[
\ln(3^{x+1}) = \ln(15)
\][/tex]
2. Step 2: Use the property of logarithms to bring down the exponent.
Use the logarithmic identity [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex] to move the exponent [tex]\(x+1\)[/tex] in front of the logarithm:
[tex]\[
(x+1) \cdot \ln(3) = \ln(15)
\][/tex]
3. Step 3: Isolate [tex]\(x+1\)[/tex].
To solve for [tex]\(x\)[/tex], divide both sides of the equation by [tex]\(\ln(3)\)[/tex]:
[tex]\[
x+1 = \frac{\ln(15)}{\ln(3)}
\][/tex]
4. Step 4: Solve for [tex]\(x\)[/tex].
Subtract 1 from both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{\ln(15)}{\ln(3)} - 1
\][/tex]
5. Step 5: Calculate the values of the logarithms.
Using the known result of the calculations (which we've verified previously):
- [tex]\(\frac{\ln(15)}{\ln(3)} \approx 2.464973520717927\)[/tex]
- Subtracting 1 from this value gives:
[tex]\[
x \approx 2.464973520717927 - 1 = 1.464973520717927
\][/tex]
Thus the value of [tex]\(x\)[/tex] is approximately [tex]\(1.464973\)[/tex]. Among the provided options, the correct one is:
1.464973
