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.