To solve the equation $5 + \log_2(-5x) = 9$, let's follow these steps:

Step 1: Isolate the logarithmic term

$\log_2(-5x) = 9 - 5$ $\log_2(-5x) = 4$

Step 2: Rewrite the equation in exponential form

The logarithmic property $\log_b(y) = x$ implies $b^x = y$. Here, $b = 2$, $x = 4$, and $y = -5x$. $2^4 = -5x$ $16 = -5x$

Step 3: Solve for $x$

$x = -\frac{16}{5}$ $x = -3.2$

Step 4: Verify the solution

The argument of the logarithm, $-5x$, must be positive for $\log_2(-5x)$ to be defined: $-5x = -5(-3.2) = 16 > 0$ The condition is satisfied, so $x = -3.2$ is valid.

Final Answer:

$\boxed{x = -3.2}$

Do you want me to elaborate on any step?

Here are 5 related questions to explore:

1. How do you convert logarithmic equations into exponential form?
2. What are the domain restrictions for logarithmic functions?
3. Can you solve $\log_b(y) = c$ when $y$ is not positive?
4. How do you verify solutions for logarithmic equations?
5. What are some real-life applications of logarithmic equations?

Tip: Always check the domain of the logarithmic function to ensure solutions are valid.