Solve the pair of simultaneous equations
\log(x + y) = 0 ...(1)
2 \log x = \log (y - 1) ...(2)
SOLUTION:
\log (x+y)=0 in index form becomes x+y = 10^0 and so
x+y=1 (1)
In (2) we see that the domain of x and y is restricted by x>0 and y-1>0 (since the log function has domain positive numbers only)
so we must have x>0 and y>1.
Now using log laws on (2) and converting to index form we have
\log (x^2) = \log(y-1)
\log(x^2) - \log(y-1) = 0
\log( x^2/(y-1) ) = 0
x^2/(y-1) = 10^0 (index form of previous line)
x^2/(y-1) = 1
x^2 = y - 1
y = x^2 + 1 (2)
So we now have the two simultaneous equations to solve
x+y=1 (1)
y = x^2 + 1 (2)
(but keeping in mind that x>0, y>1 must hold also)
so, x + x^2 +1 = 1
x^2 + x = 0
x ( x + 1 )= 0
so this gives x = 0, -1
but we must have x>0 and both of these solutions fail that condition.
So, we conclude that there are NO SOLUTIONS!
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