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Solve for [tex][tex]$x$[/tex][/tex]. Round to the nearest tenth when necessary.

[tex](1.5)^{x-1} = 14.5[/tex]

Answer: [tex]\square[/tex]

Sagot :

GinnyAnswer
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].
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