Solve the logarithmic equation.  
log left parenthesis 7 minus 2 x right parenthesis equals 2
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
xequals
  
enter your response here ​(Simplify your answer. Use a comma to separate answers as​ needed.)
B.
There is no solution.

To solve the logarithmic equation \( \log(7 - 2x) = 2 \), follow these steps:

1. **Understand the equation**: The equation \( \log(7 - 2x) = 2 \) is in the logarithmic form. To remove the logarithm, convert it to its equivalent exponential form.

2. **Convert to exponential form**: Recall that if \( \log_b(A) = C \), then \( A = b^C \). Since the base of the logarithm is 10 (common logarithm), we have:
   \[
   7 - 2x = 10^2
   \]

3. **Simplify the equation**: Calculate \( 10^2 \) which equals 100, so:
   \[
   7 - 2x = 100
   \]

4. **Solve for \( x \)**:
   \[
   -2x = 100 - 7
   \]
   \[
   -2x = 93
   \]
   \[
   x = -\frac{93}{2}
   \]
   \[
   x = -46.5
   \]

So, the solution is:
\[
x = -46.5
\]

Since the logarithm function is only defined for positive arguments, we should verify that substituting \( x = -46.5 \) into the argument of the logarithm does not result in a negative number. Substituting into the original argument:

\[
7 - 2(-46.5) = 7 + 93 = 100
\]

Since 100 is positive, the solution is valid.

**Correct Choice:**
A. \( x = -46.5 \)

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Do you want more details or have any questions?

Here are five related questions:
1. How do you solve a logarithmic equation when the base isn't specified?
2. What are the steps to convert from logarithmic form to exponential form?
3. How do you handle logarithmic equations with different bases?
4. What are the domain restrictions for logarithmic functions?
5. How would you solve \( \log_b(2x - 3) = \log_b(4x + 1) \)?

**Tip:** Always check the domain of the logarithmic function to ensure the argument stays positive when solving logarithmic equations.

Learn how to solve the logarithmic equation log(7 - 2x) = 2 step-by-step with detailed explanations and verification. This problem involves converting logarithmic equations to exponential form and solving for the variable. Suitable for high school students.

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