I simply canNOT figure out how to solve this equation: log 8x - log (1+ sqrt(x) )=2. I know the answer is 180.384 but I always get stuck when I get to figuring out what to do with the square root. Any help?
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log 8x - log (1+ sqrt(x) )=2
=> log (8x / (1 + sqrt(x)) = 2
=> 10^log (8x / (1 + sqrt(x)) = 10^2
=> 8x / (1 + sqrt(x)) = 100
=> 8x = 100(1 + sqrt(x))
=> 8x - 100 sqrt(x) - 100 = 0
now solve the quadratic for sqrt(x) using the quadratic formula:
sqrt(x) = (100 +/- sqrt(100^2 + 3200))/16
= (100 +/- sqrt(13200))/16
discard the minus because it gives a negative result, and square roots can't be negative:
=> sqrt(x) = (100 + sqrt(100^2 + 3200))/16
... then square that to get x
=> log (8x / (1 + sqrt(x)) = 2
=> 10^log (8x / (1 + sqrt(x)) = 10^2
=> 8x / (1 + sqrt(x)) = 100
=> 8x = 100(1 + sqrt(x))
=> 8x - 100 sqrt(x) - 100 = 0
now solve the quadratic for sqrt(x) using the quadratic formula:
sqrt(x) = (100 +/- sqrt(100^2 + 3200))/16
= (100 +/- sqrt(13200))/16
discard the minus because it gives a negative result, and square roots can't be negative:
=> sqrt(x) = (100 + sqrt(100^2 + 3200))/16
... then square that to get x