To solve the equation

$\frac{3117}{5 + e^{5x}} = 3$

accurately to three decimal places for $x$, let's work through the steps:

Step 1: Eliminate the denominator

Multiply through by $5 + e^{5x}$ to get: $3117 = 3(5 + e^{5x})$

Simplify: $3117 = 15 + 3e^{5x}$

Step 2: Isolate $e^{5x}$

$3117 - 15 = 3e^{5x}$

$3102 = 3e^{5x}$

$e^{5x} = \frac{3102}{3} = 1034$

Step 3: Take the natural logarithm

Take the natural logarithm of both sides to solve for $5x$: $5x = \ln(1034)$

Step 4: Solve for $x$

$x = \frac{\ln(1034)}{5}$

Using a calculator: $\ln(1034) \approx 6.942$

$x = \frac{6.942}{5} \approx 1.388$

Final Answer:

$x = 1.388$

Let me know if you'd like further clarification or detailed steps!

5 Related Questions:

1. How is the natural logarithm used in solving exponential equations?
2. What are other methods for solving equations involving exponential terms?
3. How can logarithmic identities simplify equations like this one?
4. Why is it important to isolate the exponential term first?
5. How do rounding errors affect the final answer in equations like this?

Tip: Always double-check intermediate steps to avoid small errors, especially when working with logarithmic and exponential functions.