13. \( 5(7)^{5 x}=60 \)

To solve the equation \( 5 \cdot 7^{5x} = 60 \) for \( x \), follow these steps: 1. **Isolate the Exponential Term:** Start by dividing both sides of the equation by 5 to isolate the exponential expression: \[ 7^{5x} = \frac{60}{5} = 12 \] 2. **Apply the Natural Logarithm:** Take the natural logarithm (\( \ln \)) of both sides to bring down the exponent: \[ \ln(7^{5x}) = \ln(12) \] 3. **Use Logarithm Power Rule:** The power rule of logarithms states that \( \ln(a^b) = b \cdot \ln(a) \). Apply this to the left side: \[ 5x \cdot \ln(7) = \ln(12) \] 4. **Solve for \( x \):** Divide both sides by \( 5 \ln(7) \) to isolate \( x \): \[ x = \frac{\ln(12)}{5 \ln(7)} \] 5. **(Optional) Numerical Approximation:** If you need a numerical value for \( x \), you can use a calculator to compute the natural logarithms: \[ x \approx \frac{\ln(12)}{5 \ln(7)} \approx \frac{2.4849}{5 \times 1.9459} \approx \frac{2.4849}{9.7295} \approx 0.255 \] So, \( x \approx 0.255 \). **Final Answer:** \[ x = \frac{\ln(12)}{5\, \ln(7)} \]