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Given the equation:
\[
\log_2(-5x + 7) = 1
\]

We can solve it by converting the logarithmic equation into its exponential form. Here's the step-by-step solution:

### Step 1: Rewrite the Equation in Exponential Form
The equation \(\log_2(-5x + 7) = 1\) can be rewritten as:
\[
-5x + 7 = 2^1
\]
\[
-5x + 7 = 2
\]

### Step 2: Solve for \(x\)
Rearrange the equation to isolate \(x\):
\[
-5x = 2 - 7
\]
\[
-5x = -5
\]
\[
x = 1
\]

### Step 3: Verify the Solution
Substitute \(x = 1\) back into the original equation to verify:
\[
\log_2(-5(1) + 7) = \log_2(2) = 1
\]
The solution satisfies the equation.

### Solution
The value of \(x\) is:
\[
x = 1
\]

Would you like more details on this process, or do you have any questions?

Here are five related questions for further exploration:
1. What are the general properties of logarithmic equations?
2. How do you handle logarithmic equations with a different base?
3. What are the conditions for the existence of a logarithmic expression?
4. How do you solve logarithmic equations that have more complex expressions?
5. Can you explain how to convert exponential equations back into logarithmic form?

**Tip:** Always check the argument of the logarithm to ensure it stays positive, as logarithms are only defined for positive numbers.

This page provides a step-by-step solution for solving the logarithmic equation log_2(-5x + 7) = 1. Learn how to rewrite the equation in exponential form and solve for x, applying logarithmic properties. Suitable for high school students in Grades 9-12.

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