If log (5x-15)-log (2-3) >log (2-2), then the number of positive integer values of x is:

Let's solve the inequality step by step.

We are given the inequality:

\[
\log(5x - 15) - \log(2 - 3) > \log(2 - 2)
\]

### Step 1: Simplify the logarithms
- \( \log(2 - 3) = \log(-1) \) does not exist because the logarithm of a negative number is undefined in the real number system. Therefore, the inequality has no valid solution since the logarithm is undefined in this case.

Thus, there are **no positive integer values of \( x \)** that satisfy this inequality.

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Would you like to discuss the implications of this result, or do you have any other questions?

Here are five related questions to further your understanding:
1. How do we handle logarithmic equations with negative arguments?
2. What are the domain restrictions for logarithmic functions?
3. How do we solve inequalities involving logarithms when all arguments are defined?
4. What is the difference between solving logarithmic inequalities and logarithmic equations?
5. How does the base of a logarithm affect the solution to inequalities?

**Tip:** Always check the domain of logarithmic functions before attempting to solve the equation or inequality.

Understand how to solve logarithmic inequalities using a detailed step-by-step explanation. This problem involves handling logarithms, identifying domain restrictions, and understanding when solutions exist. Suitable for advanced high school students in Grades 11-12.

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