What is the answer to (e^x+e^{-x})/(e^x-e^{-x)}=6 ?

The answer to (e^x+e^{-x})/(e^x-e^{-x)}=6 is x= 1/2 ln(7/5)

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(e^x+e^{-x})/(e^x-e^{-x)}=6

\frac{e ^{x} + e ^{- x}}{e ^{x} - e ^{- x}} = 6

x = \frac{1}{2} ln (\frac{7}{5})

Decimal Notation

x = 0.16823 \dots

Solution steps

\frac{e ^{x} + e ^{- x}}{e ^{x} - e ^{- x}} = 6

Multiply both sides by

e^{x} - e^{- x}

\frac{e ^{x} + e ^{- x}}{e ^{x} - e ^{- x}} (e^{x} - e^{- x}) = 6 (e^{x} - e^{- x})

Simplify

e^{x} + e^{- x} = 6 (e^{x} - e^{- x})

Apply exponent rules

e^{x} + (e^{x})^{- 1} = 6 (e^{x} - (e^{x})^{- 1})

Rewrite the equation with

e^{x} = u

u + (u)^{- 1} = 6 (u - (u)^{- 1})

Solve

u + u^{- 1} = 6 (u - u^{- 1}) : u = \frac{7}{5}, u = - \frac{7}{5}

u = \frac{7}{5}, u = - \frac{7}{5}

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = \frac{7}{5} : x = \frac{1}{2} ln (\frac{7}{5})

Solve

e^{x} = - \frac{7}{5} :

No Solution for

x \in R

x = \frac{1}{2} ln (\frac{7}{5})

Verify Solutions:

x = \frac{1}{2} ln (\frac{7}{5})

True

The solution is

x = \frac{1}{2} ln (\frac{7}{5})

4^{x + 1} + 2^{x + 1} + 2^{x} = 1

5^{- 2 x} = \frac{1}{25}

e^{e^{x}} = 0

x, e^{2 x} = 0

6 e^{2 x} - 11 e^{x} = - 4

What is the answer to (e^x+e^{-x})/(e^x-e^{-x)}=6 ?
The answer to (e^x+e^{-x})/(e^x-e^{-x)}=6 is x= 1/2 ln(7/5)