Which of the following is/are true?

Options

A. `log_(2)6ltlog_((1)/(2))"(1)/(5)`
B. `log_(2)5gtlog_((1)/(2))7`
C. `log_(10)12gtlog_(15)13`
D. `log_((1)/(2))3gtlog_((1)/(2))5`

Explanation

To solve the problem, we need to analyze each of the given options based on the properties of logarithms. Let's go through the steps systematically.

### Step 1: Understand the logarithmic properties
We will use the following logarithmic properties:
1. \( \log_a (b^m) = m \cdot \log_a b \)
2. If \( a > 1 \) and \( b > c \), then \( \log_a b > \log_a c \).
3. If \( 0 < a < 1 \) and \( b > c \), then \( \log_a b < \log_a c \).

### Step 2: Analyze each option

#### Option 1: \( \log_{1/2} 6 < \log_{1/5} 5 \)
- Rewrite \( \log_{1/2} 6 \) as \( -\log_2 6 \) and \( \log_{1/5} 5 \) as \( -\log_5 5 = -1 \).
- Now we have \( -\log_2 6 < -1 \) which simplifies to \( \log_2 6 > 1 \).
- Since \( 2^1 = 2 < 6 \), this is true. Thus, this option is **incorrect**.

#### Option 2: \( \log_{1/2} 1/2 < \log_2 7 \)
- Rewrite \( \log_{1/2} 1/2 \) as \( -\log_2 (1/2) = 1 \) and \( \log_2 7 \) is positive.
- Thus, \( 1 < \log_2 7 \) is true, making this option **correct**.

#### Option 3: \( \log_{1/5} 5 > \log_5 3 \)
- Rewrite \( \log_{1/5} 5 \) as \( -1 \) and \( \log_5 3 \) is positive but less than 1.
- Thus, \( -1 > \log_5 3 \) is **incorrect**.

#### Option 4: \( \log_{10} 12 > \log_{15} 13 \)
- Let \( x = \log_{10} 12 \) and \( y = \log_{15} 13 \).
- Since \( 10^x = 12 \) implies \( x > 1 \) and \( 15^y = 13 \) implies \( y < 1 \).
- Thus, \( x > 1 > y \) is true, making this option **correct**.

### Conclusion
The correct options are **2 and 4**.