To solve the equation [tex]\((1.5)^{x-1} = 14.5\)[/tex], we will use logarithms to isolate [tex]\(x\)[/tex]. Here’s the step-by-step process:
1. Start with the given equation:
[tex]\[
(1.5)^{x-1} = 14.5
\][/tex]
2. Take the natural logarithm of both sides:
[tex]\[
\ln \left( (1.5)^{x-1} \right) = \ln(14.5)
\][/tex]
3. Use the logarithm power rule [tex]\(\ln(a^b) = b \ln(a)\)[/tex] to simplify the left side:
[tex]\[
(x-1) \ln(1.5) = \ln(14.5)
\][/tex]
4. Calculate the natural logarithms:
[tex]\[
\ln(1.5) \approx 0.4055
\][/tex]
[tex]\[
\ln(14.5) \approx 2.6741
\][/tex]
5. Solve for [tex]\(x-1\)[/tex] by isolating the term:
[tex]\[
x-1 = \frac{\ln(14.5)}{\ln(1.5)}
\][/tex]
[tex]\[
x-1 \approx \frac{2.6741}{0.4055} \approx 6.5953
\][/tex]
6. Add 1 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 6.5953 + 1 \approx 7.5953
\][/tex]
7. Round the result to the nearest tenth:
[tex]\[
x \approx 7.6
\][/tex]
Therefore, the value of [tex]\(x\)[/tex], rounded to the nearest tenth, is [tex]\( \boxed{7.6} \)[/tex].
