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].
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].
