If `A=(log_(3) 2187)/5` and `B = log_(243) 2187`, then which of the following is correct?

Options

A. `A lt B`
B. `A =B`
C. `A gt B`
D. can't be determined

Answer

To solve the problem, we need to find the values of \( A \) and \( B \) and then compare them.

Given:
- \( A = \frac{\log_{3} 2187}{5} \)
- \( B = \log_{243} 2187 \)

### Step 1: Express \( 2187 \) in terms of base \( 3 \)

First, we can express \( 2187 \) as a power of \( 3 \):
\[ 2187 = 3^7 \]

### Step 2: Substitute \( 2187 \) in \( A \)

Now we substitute \( 2187 \) into \( A \):
\[ A = \frac{\log_{3} (3^7)}{5} \]

### Step 3: Simplify \( A \)

Using the logarithmic identity \( \log_{b} (b^x) = x \):
\[ A = \frac{7}{5} \]

### Step 4: Express \( 243 \) in terms of base \( 3 \)

Next, we express \( 243 \) as a power of \( 3 \):
\[ 243 = 3^5 \]

### Step 5: Substitute \( 243 \) in \( B \)

Now we can substitute \( 243 \) into \( B \):
\[ B = \log_{3^5} 2187 \]

### Step 6: Use the change of base formula for \( B \)

Using the change of base formula:
\[ B = \frac{\log_{3} 2187}{\log_{3} 243} \]

### Step 7: Substitute values into \( B \)

We already know:
- \( \log_{3} 2187 = 7 \)
- \( \log_{3} 243 = 5 \)

So we can substitute these values:
\[ B = \frac{7}{5} \]

### Step 8: Compare \( A \) and \( B \)

Now we have:
- \( A = \frac{7}{5} \)
- \( B = \frac{7}{5} \)

Thus, we can conclude:
\[ A = B \]

### Final Answer

The correct option is that \( A \) is equal to \( B \).

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