What is the inverse of f(x)=ln(((e^x))/(e^x-1)) ?

The inverse of f(x)=ln(((e^x))/(e^x-1)) is x-ln(e^x-1)

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inverse of f(x)=ln(((e^x))/(e^x-1))

inverse

f (x) = ln (\frac{( e ^{x} )}{e ^{x} - 1})

x - ln (e^{x} - 1)

Solution steps

y = ln (\frac{e ^{x}}{e ^{x} - 1})

Replace

x

with

y

x = ln (\frac{e ^{y}}{e ^{y} - 1})

Solve for

y, x = ln (\frac{e ^{y}}{e ^{y} - 1})

x - ln (e^{x} - 1)

What is the inverse of f(x)=ln(((e^x))/(e^x-1)) ?
The inverse of f(x)=ln(((e^x))/(e^x-1)) is x-ln(e^x-1)