To solve the exponential equation [tex]\(9 \cdot 5^{x-2} = 4\)[/tex], we need to isolate [tex]\(x\)[/tex]. Here is a step-by-step breakdown of the process:
1. Start with the given equation:
[tex]\[
9 \cdot 5^{x-2} = 4
\][/tex]
2. Divide both sides of the equation by 9 to isolate the exponential term:
[tex]\[
5^{x-2} = \frac{4}{9}
\][/tex]
3. To solve for [tex]\(x\)[/tex], take the natural logarithm (log base [tex]\(e\)[/tex], denoted as [tex]\(\ln\)[/tex]) of both sides. This helps to bring down the exponent:
[tex]\[
\ln\left(5^{x-2}\right) = \ln\left(\frac{4}{9}\right)
\][/tex]
4. Apply the logarithm power rule on the left-hand side, which states [tex]\(\ln(a^b) = b \ln(a)\)[/tex]:
[tex]\[
(x-2) \ln(5) = \ln\left(\frac{4}{9}\right)
\][/tex]
5. Solve for [tex]\(x-2\)[/tex] by dividing both sides by [tex]\(\ln(5)\)[/tex]:
[tex]\[
x-2 = \frac{\ln\left(\frac{4}{9}\right)}{\ln(5)}
\][/tex]
6. Rearrange to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{\ln\left(\frac{4}{9}\right)}{\ln(5)} + 2
\][/tex]
Thus, the exact solution is:
[tex]\[
x = \frac{\ln\left(4/9\right)}{\ln(5)} + 2
\][/tex]
Rewriting the logarithmic term, we get:
[tex]\[
x = \log \left(\frac{2^2}{3^2}\right)/\log(5) + 2
\][/tex]
[tex]\[
x = \log \left(2^2\right)/\log(5) - \log \left(3^2\right)/\log(5) + 2
\][/tex]
[tex]\[
x = 2 \cdot (\log(2)/\log(5)) - 2 \cdot (\log(3)/\log(5)) + 2
\][/tex]
So, the exact solution for this equation is:
[tex]\[
x = \log \left(\frac{2^2}{3^2}\right)/\log(5) + 2
\][/tex]
The numerical approximations for the logarithms were computed, resulting in an approximate solution for [tex]\(x\)[/tex]:
[tex]\[
x \approx 1.496
\][/tex]
Therefore:
1. Exact solution: [tex]\(\boxed{\log \left(\frac{2^2}{3^2}\right)/\log(5) + 2}\)[/tex]
2. Approximation: [tex]\(\boxed{1.496}\)[/tex]
