# Given that `log_b(a^2) = 3` , the value of `log_a(b^2)` is; (a) 5/3 (b) 3/4 (c) 2/3 (d) 4/3 (e) 3/2

### 2 Answers | Add Yours

`log_b(a^2) = 3`

Remove the logarithm will give us;

`a^2 = b^3`

`b = (a^2)^(1/3)`

`b^2 = a^(4/3)`

Take log on both sides using a as base.

`log_a(b^2) = log_a(a^(4/3))`

`log_a(b^2) = 4/3log_a(a)`

`log_a(b^2) = 4/3`

*So the correct answer is at option d)*

**Sources:**

If `log_b x = y` , `x = b^y` .

In the problem it is given that `log_b a^2 = 3` .

This gives `a^2 = b^3`

Now take the log to base a of both the sides

`log_a a^2 = log_a b^3`

Use the relation `log a^b = b*log a` and `log_a a = 1`

`log_a a^2 = log_a b^3`

`2*log_a a = 3*log_a b`

`2 = 3*log_a b`

Multiply both sides by 2/3

`4/3 = 2*log_a b`

`4/3 = log_a b^2`

The value of `log_a b^2 = 4/3`