We are given the equation:

$\log_e(8x + 6) = -1$

This is a logarithmic equation, and $\log_e$ is the natural logarithm, often denoted as $\ln$. To solve for $x$, follow these steps:

Step 1: Rewrite the equation in exponential form

The logarithmic equation can be rewritten as an exponential equation. The general rule is:

$\log_b(a) = c \implies a = b^c$

In our case:

$\ln(8x + 6) = -1 \implies 8x + 6 = e^{-1}$

Step 2: Simplify the exponential

We know that $e^{-1} = \frac{1}{e}$. Therefore, the equation becomes:

$8x + 6 = \frac{1}{e}$

Step 3: Solve for $x$

Now, isolate $x$ by following these steps:

1. Subtract 6 from both sides: $8x = \frac{1}{e} - 6$

2. Simplify the right-hand side: $8x = \frac{1}{e} - \frac{6e}{e} = \frac{1 - 6e}{e}$

3. Divide both sides by 8: $x = \frac{1 - 6e}{8e}$

Thus, the solution is:

$x = \frac{1 - 6e}{8e}$

Would you like more details on any of the steps?

5 related questions:

1. How do you convert logarithmic equations into exponential form?
2. What are the properties of natural logarithms and their uses in solving equations?
3. Can you explain how the base of natural logarithms, $e$, is derived?
4. What are common methods for solving logarithmic equations with multiple terms?
5. How do you simplify exponential expressions involving $e$?

Tip: Always check if the logarithmic expression's argument is positive, as the logarithm is only defined for positive numbers.