Question

Solve for x.

2 log₁₀ x = `1 + log_10 (x + 11/10)`

Answer 1

2 log₁₀ x = `1 + log_10 (x + 11/10)`

∴ `log_10 x^2 - log_10 (x + 11/10)` = 1 ...[n log m = log mⁿ]

∴ `log_10 x^2 - log_10((10x + 11)/10)` = 1

∴ `log_10 (x^2/((10 x + 11)/10))` = 1  ...`[log"m" - log "n" = log  "m"/"n"]`

∴ `log_10 ((10x^2)/(10x + 11))` = log₁₀ 10   ...[∵ log_a a = 1]

∴ `(10x^2)/(10x + 11)` = 10 

∴  `x^2/(10x + 11)` = 1

∴ x² = 10x + 11

∴ x²  – 10x – 11 = 0

∴ (x – 11)(x + 1) = 0

∴ x = 11 or x = – 1

But log of negative number does not exist

∴ x ≠ – 1

∴ x = 11