CAT 2003 Slot 2 — QA Question 43

log₁₀ x - log₁₀ $\sqrt{x}$ = 2 log_x 10 ....(i)

Consider L.H.S of (i):

log₁₀ x - $\frac{1}{2}$ log₁₀ x = $\frac{1}{2}$ log₁₀ x

Equating the L.H.S. and the R.H.S., we get,

log₁₀ x = 4 log_x 10

∴ log₁₀ x = 4 log_x 10⁴

Now, let log₁₀ x = log_x 10⁴ = y

∴ 10^y = x and x^y = 10⁴

Solving these two equations, we get,

y = ±2

∴ x = 10² or x = 10^–2

As x = 100 is not in the options,

∴ x = 10⁻² = 1/100

Hence, option (b).