The value of `x` in the expression `(x+x^((log)_(10)))^5` if third term in the expansion is 10,00,000 is/are a. 10 b. 100 c. `10^(-5//2)` d. `10^(-3//2)`

Options

A. 10
B. 100
C. `10^(-5//2)`
D. `10^(-3//2)`

Answer

A, C

Explanation

Inclusion of log x implies `x gt 0`
Now, `3^(nd)` term in expension is
`T_(2+1) = .^(5)C_(2)x^(5-2)(x^(log_(10)x))^(2) = 10000000` (given)
or `x^(3+2log_(10)x) = 10^(5)`
Taking logarithm of both sides, we get
`(3+2log_(10)x)log_(10)x = 5`
or `2y^(2) + 3 y - 5 = 0`, where `log_(10) x = y`
or `(y-1) (2y+5) = 0 y = 1` or `-5//2`
or `log_(10)x=1` or `-5//2`
`:. x = 10^(1) = 10` or `10^(-5//2)`