If `A= log_(13) 189` and `B = log_(23)521`, then which one of the following is correct?

Options

A. `A lt B`
B. `A =B`
C. `A gt B`
D. none of these

Answer

To solve the problem, we need to analyze the logarithmic values of \( A \) and \( B \) given by:

\[ A = \log_{13}(189) \]
\[ B = \log_{23}(521) \]

### Step 1: Analyze \( A \)

We start with \( A = \log_{13}(189) \). We can express \( 189 \) in terms of \( 13 \):

\[ 189 = 13^2 + 20 \]

Now, we know that \( 13^2 = 169 \). Therefore, we can write:

\[ 189 = 169 + 20 \]

This means \( 189 \) is slightly greater than \( 13^2 \).

Using the properties of logarithms, we can say:

\[ A = \log_{13}(13^2 + 20) \]

Since \( 13^2 = 169 \), we can conclude that:

\[ A > 2 \]

### Step 2: Analyze \( B \)

Next, we analyze \( B = \log_{23}(521) \). We can express \( 521 \) in terms of \( 23 \):

\[ 23^2 = 529 \]

Thus, we can write:

\[ 521 = 529 - 8 \]

Using the properties of logarithms, we can say:

\[ B = \log_{23}(529 - 8) \]

Since \( 521 \) is slightly less than \( 23^2 \), we can conclude that:

\[ B < 2 \]

### Step 3: Compare \( A \) and \( B \)

From our analysis, we have:

\[ A > 2 \quad \text{and} \quad B < 2 \]

This implies that:

\[ A > B \]

### Conclusion

Thus, the correct statement is that \( A \) is greater than \( B \).

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