What is the inverse of f(x)=5^x-5^{-x} ?

The inverse of f(x)=5^x-5^{-x} is (ln((x+sqrt(x^2+4))/2))/(ln(5)),(ln((x-sqrt(x^2+4))/2))/(ln(5))

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inverse of f(x)=5^x-5^{-x}

inverse

f (x) = 5^{x} - 5^{- x}

\frac{ln ( \frac{x + x ^{2} + 4}{2} )}{ln ( 5 )}, \frac{ln ( \frac{x - x ^{2} + 4}{2} )}{ln ( 5 )}

Solution steps

y = 5^{x} - 5^{- x}

Replace

x

with

y

x = 5^{y} - 5^{- y}

Solve for

y, x = 5^{y} - 5^{- y}

\frac{ln ( \frac{x + x ^{2} + 4}{2} )}{ln ( 5 )}, \frac{ln ( \frac{x - x ^{2} + 4}{2} )}{ln ( 5 )}

What is the inverse of f(x)=5^x-5^{-x} ?
The inverse of f(x)=5^x-5^{-x} is (ln((x+sqrt(x^2+4))/2))/(ln(5)),(ln((x-sqrt(x^2+4))/2))/(ln(5))