The equation to solve is:

$8 + \log_5(-4x) = 10$

Step 1: Isolate the logarithmic term

Subtract 8 from both sides:

$\log_5(-4x) = 10 - 8$

$\log_5(-4x) = 2$

Step 2: Rewrite in exponential form

The property of logarithms $\log_b(a) = c \implies b^c = a$ allows us to rewrite:

$5^2 = -4x$

$25 = -4x$

Step 3: Solve for $x$

Divide both sides by $-4$:

$x = \frac{25}{-4}$

$x = -\frac{25}{4}$

Step 4: Verify the solution

For logarithms to be valid, the argument must be positive. Here, the argument is $-4x$:

$-4(-\frac{25}{4}) = 25$

Since the argument is positive, the solution is valid.

Final Answer:

$x = -\frac{25}{4}$

Let me know if you’d like further clarification!

Related Questions:

1. What are the rules for solving logarithmic equations with different bases?
2. How do you rewrite logarithmic equations in exponential form?
3. What happens if the logarithmic argument is negative or zero?
4. Can logarithmic equations have multiple solutions?
5. How do you evaluate logarithms without a calculator?

Tip:

Always verify that the argument of the logarithm is positive after solving to ensure the solution is valid!