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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:
\[
\boxed{x = -3.2}
\]

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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.

Explore a step-by-step solution for the logarithmic equation 5 + log_2(-5x) = 9. Learn key concepts like logarithmic properties, domain restrictions, and solving exponential equations. Suitable for high school students in Grades 10-12.

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