What is the general solution for tanh(x)=0.99 ?

The general solution for tanh(x)=0.99 is x= 1/2 ln(199)

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tanh(x)=0.99

Solution

tanh (x) = 0.99

Solution

x = \frac{1}{2} ln (199)

Degrees

x = 151.64201 \dots^{\circ}

Solution steps

tanh (x) = 0.99

Rewrite using trig identities

tanh (x) = 0.99

Use the Hyperbolic identity:

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

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 0.99

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 0.99

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 0.99 : x = \frac{1}{2} ln (199)

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 0.99

Multiply both sides by

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

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} (e^{x} + e^{- x}) = 0.99 (e^{x} + e^{- x})

Simplify

e^{x} - e^{- x} = 0.99 (e^{x} + e^{- x})

Apply exponent rules

e^{x} - e^{- x} = 0.99 (e^{x} + e^{- x})

Apply exponent rule:

a^{b c} = (a^{b})^{c}

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

e^{x} - (e^{x})^{- 1} = 0.99 (e^{x} + (e^{x})^{- 1})

e^{x} - (e^{x})^{- 1} = 0.99 (e^{x} + (e^{x})^{- 1})

Rewrite the equation with

e^{x} = u

u - (u)^{- 1} = 0.99 (u + (u)^{- 1})

Solve

u - u^{- 1} = 0.99 (u + u^{- 1}) : u = 199, u = - 199

u - u^{- 1} = 0.99 (u + u^{- 1})

Refine

u - \frac{1}{u} = 0.99 (u + \frac{1}{u})

Multiply both sides by

100

u - \frac{1}{u} = 0.99 (u + \frac{1}{u})

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

u \cdot 100 - \frac{1}{u} \cdot 100 = 0.99 (u + \frac{1}{u}) \cdot 100

Refine

100 u - \frac{100}{u} = 99 (u + \frac{1}{u})

100 u - \frac{100}{u} = 99 (u + \frac{1}{u})

Multiply both sides by

u

100 u - \frac{100}{u} = 99 (u + \frac{1}{u})

Multiply both sides by

u

100 uu - \frac{100}{u} u = 99 (u + \frac{1}{u}) u

Simplify

100 uu - \frac{100}{u} u = 99 (u + \frac{1}{u}) u

Simplify

100 uu : 100 u^{2}

100 uu

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= 100 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 100 u^{2}

Simplify

- \frac{100}{u} u : - 100

- \frac{100}{u} u

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{100 u}{u}

Cancel the common factor:

u

= - 100

100 u^{2} - 100 = 99 (u + \frac{1}{u}) u

100 u^{2} - 100 = 99 (u + \frac{1}{u}) u

100 u^{2} - 100 = 99 (u + \frac{1}{u}) u

Expand

99 (u + \frac{1}{u}) u : 99 u^{2} + 99

99 (u + \frac{1}{u}) u

= 99 u (u + \frac{1}{u})

Apply the distributive law:

a (b + c) = ab + a c

a = 99 u, b = u, c = \frac{1}{u}

= 99 uu + 99 u \frac{1}{u}

= 99 uu + 99 \cdot \frac{1}{u} u

Simplify

99 uu + 99 \cdot \frac{1}{u} u : 99 u^{2} + 99

99 uu + 99 \cdot \frac{1}{u} u

99 uu = 99 u^{2}

99 uu

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= 99 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 99 u^{2}

99 \cdot \frac{1}{u} u = 99

99 \cdot \frac{1}{u} u

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 99 u}{u}

Cancel the common factor:

u

= 1 \cdot 99

Multiply the numbers:

1 \cdot 99 = 99

= 99

= 99 u^{2} + 99

= 99 u^{2} + 99

100 u^{2} - 100 = 99 u^{2} + 99

Move

100

to the right side

100 u^{2} - 100 = 99 u^{2} + 99

Add

100

to both sides

100 u^{2} - 100 + 100 = 99 u^{2} + 99 + 100

Simplify

100 u^{2} = 99 u^{2} + 199

100 u^{2} = 99 u^{2} + 199

Solve

100 u^{2} = 99 u^{2} + 199 : u = 199, u = - 199

100 u^{2} = 99 u^{2} + 199

Move

99 u^{2}

to the left side

100 u^{2} = 99 u^{2} + 199

Subtract

99 u^{2}

from both sides

100 u^{2} - 99 u^{2} = 99 u^{2} + 199 - 99 u^{2}

Simplify

u^{2} = 199

u^{2} = 199

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 199, u = - 199

u = 199, u = - 199

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

u - u^{- 1}

and compare to zero

u = 0

Take the denominator(s) of

0.99 (u + u^{- 1})

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 199, u = - 199

u = 199, u = - 199

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = 199 : x = \frac{1}{2} ln (199)

e^{x} = 199

Apply exponent rules

e^{x} = 199

Apply exponent rule:

a = a^{\frac{1}{2}}

199 = 19 9^{\frac{1}{2}}

e^{x} = 19 9^{\frac{1}{2}}

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln (19 9^{\frac{1}{2}})

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (19 9^{\frac{1}{2}})

Apply log rule:

ln (x^{a}) = a \cdot ln (x)

ln (19 9^{\frac{1}{2}}) = \frac{1}{2} ln (199)

x = \frac{1}{2} ln (199)

x = \frac{1}{2} ln (199)

Solve

e^{x} = - 199 :

No Solution for

x \in R

e^{x} = - 199

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = \frac{1}{2} ln (199)

Verify Solutions:

x = \frac{1}{2} ln (199)

True

Check the solutions by plugging them into

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 0.99

Remove the ones that don't agree with the equation.

Plug in

x = \frac{1}{2} ln (199) :

True

\frac{e ^{\frac{1}{2} l n (199)} - e ^{- \frac{1}{2} l n (199)}}{e ^{\frac{1}{2} l n (199)} + e ^{- \frac{1}{2} l n (199)}} = 0.99

\frac{e ^{\frac{1}{2} l n (199)} - e ^{- \frac{1}{2} l n (199)}}{e ^{\frac{1}{2} l n (199)} + e ^{- \frac{1}{2} l n (199)}} = \frac{99}{100}

\frac{e ^{\frac{1}{2} l n (199)} - e ^{- \frac{1}{2} l n (199)}}{e ^{\frac{1}{2} l n (199)} + e ^{- \frac{1}{2} l n (199)}}

e^{\frac{1}{2} l n (199)} = 199

e^{\frac{1}{2} l n (199)}

Apply exponent rule:

a^{b c} = (a^{b})^{c}

= e^{l n (199)}

Apply log rule:

a^{l o g_{a} (b)} = b

e^{l n (199)} = 199

= 199

e^{- \frac{1}{2} l n (199)} = 19 9^{- \frac{1}{2}}

e^{- \frac{1}{2} l n (199)}

Apply exponent rule:

a^{b c} = (a^{b})^{c}

= (e^{l n (199)})^{- \frac{1}{2}}

Apply log rule:

a^{l o g_{a} (b)} = b

e^{l n (199)} = 199

= 19 9^{- \frac{1}{2}}

= \frac{e ^{\frac{1}{2} l n (199)} - e ^{- \frac{1}{2} l n (199)}}{199 + 19 9 ^{- \frac{1}{2}}}

e^{\frac{1}{2} l n (199)} = 199

e^{\frac{1}{2} l n (199)}

Apply exponent rule:

a^{b c} = (a^{b})^{c}

= e^{l n (199)}

Apply log rule:

a^{l o g_{a} (b)} = b

e^{l n (199)} = 199

= 199

e^{- \frac{1}{2} l n (199)} = 19 9^{- \frac{1}{2}}

e^{- \frac{1}{2} l n (199)}

Apply exponent rule:

a^{b c} = (a^{b})^{c}

= (e^{l n (199)})^{- \frac{1}{2}}

Apply log rule:

a^{l o g_{a} (b)} = b

e^{l n (199)} = 199

= 19 9^{- \frac{1}{2}}

= \frac{199 - 19 9 ^{- \frac{1}{2}}}{199 + 19 9 ^{- \frac{1}{2}}}

Simplify

\frac{199 - 19 9 ^{- \frac{1}{2}}}{199 + 19 9 ^{- \frac{1}{2}}}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

19 9^{- \frac{1}{2}} = \frac{1}{199}

= \frac{199 - 19 9 ^{- \frac{1}{2}}}{199 + \frac{1}{199}}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

19 9^{- \frac{1}{2}} = \frac{1}{199}

= \frac{199 - \frac{1}{199}}{199 + \frac{1}{199}}

Join

199 + \frac{1}{199} : \frac{200}{199}

199 + \frac{1}{199}

Convert element to fraction:

199 = \frac{199 199}{199}

= \frac{199 199}{199} + \frac{1}{199}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{199 199 + 1}{199}

199199 + 1 = 200

199199 + 1

Apply radical rule:

a a = a

199199 = 199

= 199 + 1

Add the numbers:

199 + 1 = 200

= 200

= \frac{200}{199}

= \frac{199 - \frac{1}{199}}{\frac{200}{199}}

Join

199 - \frac{1}{199} : \frac{198}{199}

199 - \frac{1}{199}

Convert element to fraction:

199 = \frac{199 199}{199}

= \frac{199 199}{199} - \frac{1}{199}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{199 199 - 1}{199}

199199 - 1 = 198

199199 - 1

Apply radical rule:

a a = a

199199 = 199

= 199 - 1

Subtract the numbers:

199 - 1 = 198

= 198

= \frac{198}{199}

= \frac{\frac{198}{199}}{\frac{200}{199}}

Divide fractions:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

= \frac{198 199}{199 \cdot 200}

Cancel the common factor:

199

= \frac{198}{200}

Cancel the common factor:

2

= \frac{99}{100}

= \frac{99}{100}

\frac{99}{100} = 0.99

True

The solution is

x = \frac{1}{2} ln (199)

x = \frac{1}{2} ln (199)

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Frequently Asked Questions (FAQ)

What is the general solution for tanh(x)=0.99 ?
The general solution for tanh(x)=0.99 is x= 1/2 ln(199)