log(x/10) = log(10/x)+2 what is X

log(x/10) = log(10/x)+2           [   2 = log (100) ]

log(x/10)  = log (10/x) + log (100)

log(x / 10)  = log ( 10/x * 100)

log(x/10)  = log ( 1000/x )     and......equating logs, we have

x/10  = 1000/x       cross-multiply

x^2  = 10000

x = 100

Solve for x: (log(x/10))/(log(10)) = 2+(log(10/x))/(log(10)) Subtract 2+(log(10/x))/(log(10)) from both sides: -2-(log(10/x))/(log(10))+(log(x/10))/(log(10)) = 0 Bring -2-(log(10/x))/(log(10))+(log(x/10))/(log(10)) together using the common denominator log(10): -(2 log(10)+log(10/x)-log(x/10))/(log(10)) = 0 Multiply both sides by -log(10): 2 log(10)+log(10/x)-log(x/10) = 0 2 log(10)+log(10/x)-log(x/10) = log(100)+2 log(10/x) = log((100 10 10)/(x x)) = log(10000/x^2): log(10000/x^2) = 0 Cancel logarithms by taking exp of both sides: 10000/x^2 = 1 Take the reciprocal of both sides: x^2/10000 = 1 Multiply both sides by 10000: x^2 = 10000 Take the square root of both sides: Answer: | | x = 100   or   x = -100

Sorry, guest......

x cannot be -100  because it would mean that we would be taking the log of a negative number on both sides in the original equation....!!!

CPhill: I agree with you. However, the software spat that out  qualifying it thus: "x=-100 [assuming a complex-valued logarithm]", which I ignored!.

Alternatively, avoiding the x squared term,

log(x/10) = log(10/x) + 2,

log(x) - log(10) = log(10) - log(x) + 2,

2log(x) = 2log(10) + 2

= 2 + 2

= 4,

log(x) = 2,

x = 100.

LOG (x/10) =  LOG(10/x) + 2     Raise both sides to the 10

10^(log(x/10) = 10^(log(10/x) +2)  

x/10 = 100 (10/x)

x/1000 = 10/x

x^2/1000 = 10

x=100