Solve the exponential equation. Express irrational solutions as decimals correct to the nearest thousandth.

[tex]\[ 5^{x-1} = 3^{2x} \][/tex]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. The solution set is \{ \(\square\) \}.
(Round to the nearest thousandth as needed. Use a comma to separate answers as needed.)

B. The solution is the empty set.



Answer :

To solve the exponential equation \( 5^{x-1} = 3^{2x} \), let's go through the problem step-by-step:

1. Rewrite the equation in exponential form:
[tex]\[ 5^{x-1} = 3^{2x} \][/tex]

2. Take the natural logarithm (ln) on both sides to leverage the properties of logarithms to bring the exponents down:
[tex]\[ \ln(5^{x-1}) = \ln(3^{2x}) \][/tex]

3. Use the power rule of logarithms, which states that \(\ln(a^b) = b \cdot \ln(a) \):
[tex]\[ (x-1) \cdot \ln(5) = 2x \cdot \ln(3) \][/tex]

4. Distribute the logarithms:
[tex]\[ x \cdot \ln(5) - \ln(5) = 2x \cdot \ln(3) \][/tex]

5. Rearrange terms to isolate \(x\). Combine like terms together:
[tex]\[ x \cdot \ln(5) - 2x \cdot \ln(3) = \ln(5) \][/tex]

6. Factor out \(x\) from the left-hand side:
[tex]\[ x (\ln(5) - 2 \ln(3)) = \ln(5) \][/tex]

7. Solve for \(x\) by dividing both sides by \((\ln(5) - 2\ln(3))\):
[tex]\[ x = \frac{\ln(5)}{\ln(5) - 2 \ln(3)} \][/tex]

8. Substitute \(\ln(5)\) and \(\ln(3)\) with their numerical values and calculate the result. However, let's provide the answer directly rounded to the nearest thousandth:

Hence, the value of \( x \) rounded to the nearest thousandth is:
[tex]\[ x \approx -2.738 \][/tex]

Therefore, the solution set is:
[tex]\[ \{-2.738\} \][/tex]

So, the correct choice is:
A. The solution set is [tex]\(\{-2.738\}\)[/tex].