If log_a ( ab ) = x , then find log_b ( ab ) ?

Solution:

Given: log_a(ab) = x

We need to find log_b(ab).

Concept Used:

The logarithmic property states:

log_a(ab) = log_a(a) + log_a(b)

Here, log_a(a) = 1.

Thus, log_a(ab) = 1 + log_a(b).

If log_a(ab) = x, then log_a(b) = x - 1.

Using the base conversion formula:

log_b(ab) = log_a(ab) / log_a(b).

Calculation:

Given log_a(ab) = x and log_a(b) = x - 1:

log_b(ab) = log_a(ab) / log_a(b)

⇒ log_b(ab) = x / (x - 1)

Comparing with the options provided:

Option 1: x / (1 + x)

Option 2: x / (x - 1)

Option 3: 1 / x

Option 4: x / (1 - x)

Correct answer is Option 2.

Conclusion: ∴ log_b(ab) = x / (x - 1), which corresponds to Option 2.