Question

If x is a positive quantity such that 2^x = 3^log₅2, then x is equal to
1. \(1-log_5\frac{5}{3}\)
2. logs9
3. logs8
4. \(1+log_3\frac{3}{5}\)

Answer 1

Correct Answer - Option 1 : \(1-log_5\frac{5}{3}\)

Calculation:

⇒ 2^x = 3^log₅2

Taking log both sides

⇒ x log 2 = log₅2 log 3

⇒ x/log₅2 = log3/log2

⇒ x/(log2/log5) = log3/log2

⇒ x log 5 = log 3

⇒ x = log 3/log5

⇒ x = log₅3

From options 1:

⇒ 1 - log₅5/3 = 1 - log₅5 + log₅3 = log₅3            ∴log55 = 1

∴ x = 1 - log₅5/3

log_ab = (log_mb)/(log_ma)