What is the inverse of # y=2log(3x-1)-log(x)#?

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Answer 1 · 2015-12-09T15:10:47

#f^-1(x) = frac{ 10^x + 6 ± sqrt {10^x (10^x + 12)}}{18}#

We want x such that #log 10^y = log frac{(3x -1 )^2}{x}, 3x - 1 >0, x> 0#

#10^y * x = 9x^2 - 6x + 1#

#0 = 9x^2 - bx + 1 ; b = 10^y + 6#

#Delta = b^2 - 36 = 10^(2y) + 12 * 10^y#

#x = frac{b ± sqrt Delta}{18} > 1/3#

#b ± sqrt Delta > 6#

#± sqrt Delta > -10^y#