If the third in the expansion of (1/x+ x^log10x)^5 is 1000, find . ​

Question

If the third in the expansion of (\(\frac{1}{x}\)+ x^log_10x)⁵ is 1000, find x.

Answer 1

Given, T₃ = 1000 ⇒ T^{2 + 1} = 1000 

⇒ ⁵C₂\((\frac{1}{x})\) ^{5 − 2} (x ^{log x} ) ² = 1000 

⇒ 10 (x ^{log10 x} ) ² x ^{− 3} = 1000 

⇒ x ^{2 log} ₁₀ x. x ^{− 3} = 100

⇒ x^2log₁₀^{x −3} = 10³ 

⇒ 2 log₁₀ x− 3 = log_x10² 

⇒ 2y− 3 = 2/y , where y = log10 x

⇒ 2y² − 3y − 2 = 0

 ⇒ (2y + 1) ( y− 2) = 0 

⇒y = 2

⇒ log₁₀ x= 2 

⇒x = 10² = 100