To solve the equation [tex]\(5^{x-2} = 8\)[/tex] using logarithms, follow these steps:
1. Take the logarithm of both sides:
We apply the common logarithm (logarithm base 10) to both sides of the equation:
[tex]\[
\log(5^{x-2}) = \log(8)
\][/tex]
2. Use the power rule of logarithms:
The power rule states [tex]\(\log(a^b) = b \log(a)\)[/tex]. Using this rule, we can rewrite the left side:
[tex]\[
(x-2) \log(5) = \log(8)
\][/tex]
3. Solve for [tex]\(x-2\)[/tex]:
Isolate [tex]\(x-2\)[/tex] by dividing both sides of the equation by [tex]\(\log(5)\)[/tex]:
[tex]\[
x-2 = \frac{\log(8)}{\log(5)}
\][/tex]
4. Compute the values of [tex]\(\log(5)\)[/tex] and [tex]\(\log(8)\)[/tex]:
Using a calculator, we find:
[tex]\[
\log(5) \approx 0.698970
\][/tex]
[tex]\[
\log(8) \approx 0.903090
\][/tex]
5. Substitute these values back into the equation:
Plugging in the values, we get:
[tex]\[
x-2 = \frac{0.903090}{0.698970} \approx 1.292030
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
Finally, add 2 to both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
x = 1.292030 + 2 = 3.292030
\][/tex]
7. Round the answer to the nearest thousandth:
The final value of [tex]\(x\)[/tex] rounded to the nearest thousandth is:
[tex]\[
x \approx 3.292
\][/tex]
Thus, the solution to the equation [tex]\(5^{x-2} = 8\)[/tex], rounded to the nearest thousandth, is [tex]\(x \approx 3.292\)[/tex].
