Which of the following is true?

Options

A. `log_(17)275=log_(19)375`
B. `log_(17)275ltlog_(19)375`
C. `log_(17)275gtlog_(19)375`
D. None of these

Answer

To determine which of the given options is true, we need to evaluate the logarithmic expressions and compare them. The question involves comparing \( \log_{275}(17) \) and \( \log_{375}(19) \).

### Step-by-Step Solution:

**Step 1: Understand the logarithmic expressions.**
We need to compare \( \log_{275}(17) \) and \( \log_{375}(19) \).

**Step 2: Use the change of base formula.**
The change of base formula states that:
\[ \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \]
for any base \( k \). We can use base 10 or natural logarithm (base \( e \)). Let's use base 10 for simplicity.

Thus, we have:
\[ \log_{275}(17) = \frac{\log(17)}{\log(275)} \]
\[ \log_{375}(19) = \frac{\log(19)}{\log(375)} \]

**Step 3: Calculate the logarithmic values.**
Using a calculator, we can find:
- \( \log(17) \approx 1.2304 \)
- \( \log(275) \approx 2.4395 \)
- \( \log(19) \approx 1.2788 \)
- \( \log(375) \approx 2.5682 \)

Now substituting these values into our expressions:
\[ \log_{275}(17) \approx \frac{1.2304}{2.4395} \approx 0.504 \]
\[ \log_{375}(19) \approx \frac{1.2788}{2.5682} \approx 0.498 \]

**Step 4: Compare the two logarithmic values.**
Now we compare:
\[ \log_{275}(17) \approx 0.504 \quad \text{and} \quad \log_{375}(19) \approx 0.498 \]
Since \( 0.504 > 0.498 \), we conclude that:
\[ \log_{275}(17) > \log_{375}(19) \]

**Step 5: Evaluate the options.**
Based on our calculations:
- If option 1 states \( \log_{275}(17) = \log_{375}(19) \), it is false.
- If option 2 states \( \log_{275}(17) < \log_{375}(19) \), it is false.
- If option 3 states \( \log_{275}(17) > \log_{375}(19) \), it is true.
- If option 4 states something else, we can disregard it as well.

Thus, the correct answer is that option 3 is true.