Solve the equation log(1-x) - log(x) = 1 where log() is the logarithmic function, base 10.

From the rules of logarithms, we know that:

log(A) - log(B) = log(A/B)

and thus:

log(1-x) - log(x) = log[(1-x)/x]

Therefore from the question, we know:

log[(1-x)/x] = 1

If we then take both sides of the equation as a power of 10:

(1-x)/x = 10^1

and then multiply both sides through by x:


Solving for x:



We can check our answer by inserting it into the original equation:

log(1-x) - log(x) = log[1-(1/11)] - log[1/11]

and using the rule log(A) - log(B) = log(A/B):

log[1-(1/11)] - log[1/11] = log(10/11)-log(1/11)

= log[(10/11)/(1/11)]

= log(10)

= 1

Thus we know x=1/11

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