Question

Given 2 log₁₀ x + 1 = log₁₀ 250, find :
(i) x
(ii) log₁₀ 2x

Answer 1

(i) Consider the given equation : 
2log₁₀x + 1 = log₁₀250
⇒ log₁₀x² + 1 = log₁₀250     [ log_amⁿ = nlog_am]

⇒ log₁₀x2 + log₁₀10 = log₁₀250  [ ∵ log₁₀10 = 1]

⇒ log₁₀( x² x 10 ) = log₁₀250       [ log_am + log_an = log_amn ]

⇒ x² x 10 = 250
⇒ x2 = 25
⇒ x = `sqrt25`
⇒ x = 5

(ii) x = 5 ( proved above in (i))
log₁₀2x = log₁₀2(5)
= log₁₀10
= 1                               [ ∵ log₁₀10 = 1]