Question

If a^x=(a/k)^y=k^m then find 1/x-1/y​

Answer 1

Step-by-step explanation:

Given : a^x = (a/k)^y = k^m.

Apply 'log' on both sides, we get

⇒ log(a^x) = log(a/k)^y = log(k)^m

We know that log(a)^n = n log a

⇒ x log a = y log(a/k) = m log k.

We know that log(a/b) = log a - log b

⇒ x log a = y(log a - log k) = m log k

Now,

Equating like terms, we get

⇒ x = (m log k)/(log a)

⇒ y = (m log k)/(log a - log k).

Given:

⇒ (1/x) - (1/y)

⇒ (log a/m log k) - (log a - log k/m log k)

⇒ (log a - log a + log k/m log k)

⇒ (log k)/(m log k)

⇒ 1/m.

Therefore, 1/x - 1/y = 1/m.

Hope it helps!