To solve the given exponential equation:

$5^{5x + 7} = 3^{x - 2}$

Step 1: Take the natural logarithm (ln) of both sides:

$\ln(5^{5x + 7}) = \ln(3^{x - 2})$

Step 2: Use the logarithmic property $\ln(a^b) = b \cdot \ln(a)$:

$(5x + 7) \cdot \ln(5) = (x - 2) \cdot \ln(3)$

Step 3: Expand both sides:

$5x \cdot \ln(5) + 7 \cdot \ln(5) = x \cdot \ln(3) - 2 \cdot \ln(3)$

Step 4: Rearrange terms to isolate $x$:

$5x \cdot \ln(5) - x \cdot \ln(3) = -2 \cdot \ln(3) - 7 \cdot \ln(5)$

Factor $x$ on the left-hand side: $x \cdot (5 \cdot \ln(5) - \ln(3)) = -2 \cdot \ln(3) - 7 \cdot \ln(5)$

Step 5: Solve for $x$:

$x = \frac{-2 \cdot \ln(3) - 7 \cdot \ln(5)}{5 \cdot \ln(5) - \ln(3)}$

Final Answer (Exact Form):

$x = \frac{-2 \ln(3) - 7 \ln(5)}{5 \ln(5) - \ln(3)}$

Step 6: Use a calculator to approximate the solution:

1. Compute the values of $\ln(3)$ and $\ln(5)$:

   • $\ln(3) \approx 1.0986$
   • $\ln(5) \approx 1.6094$

2. Substitute into the equation: $x = \frac{-2(1.0986) - 7(1.6094)}{5(1.6094) - 1.0986}$

Simplify: $x \approx \frac{-2.1972 - 11.2658}{8.047 - 1.0986} = \frac{-13.463}{6.9484} \approx -1.938$

Final Approximation:

$x \approx -1.94$

Let me know if you need further clarification or step-by-step assistance!

5 Follow-Up Questions:

1. Would you like a detailed explanation of the logarithmic properties used?
2. Do you want to understand how to compute natural logarithms on a calculator?
3. Should we verify the solution by substitution into the original equation?
4. Would you like to explore alternative methods for solving exponential equations?
5. Are there other equations you'd like help solving?

Tip:

Always simplify logarithmic equations step-by-step and verify intermediate results to minimize errors in approximation.