Question

Please answer this question.
My doubt: In log n with base a, a>0 . Here in LHS it is given -x. How? or is x originally -ve?

Answer 1

Yes. It's given negative x, but we know that the log input must always be positive, so (-x) >0 which implies x must be negative.

Answer 2

√( log(-x)) = log√(x²)

we know , for condition of log( a base b)
a >0 and b> 0 but b ≠ 1

now, here ,
log( -x) is given, for log to be defined
-x>0 so, x < 0

we also know that √x² = |x |
but x < 0
so, √x² = -x

let ( -x ) = P
now,
√( log(-x) ) = log(√x²)

√{log( -x) = log( -x)

√logP = logP

take square both sides

logP = logP²

logP ( logP -1) = 0

logP = 0, 1

P = 1 , 10
-x = 1, 10

x = -1 , -10

also x < 0

so , x = -1, -10 both are the solution of this
hence option ( c) is correct