What is the general solution for (1-tanh(x))/(1+tanh(x))=2 ?

The general solution for (1-tanh(x))/(1+tanh(x))=2 is x=-1/2 ln(2)

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(1-tanh(x))/(1+tanh(x))=2

Solution

\frac{1 - tanh ( x )}{1 + tanh ( x )} = 2

Solution

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

Degrees

x = - 19.85720 \dots^{\circ}

Solution steps

\frac{1 - tanh ( x )}{1 + tanh ( x )} = 2

Rewrite using trig identities

\frac{1 - tanh ( x )}{1 + tanh ( x )} = 2

Use the Hyperbolic identity:

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

\frac{1 - \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}}{1 + \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}} = 2

\frac{1 - \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}}{1 + \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}} = 2

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

\frac{1 - \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}}{1 + \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}} = 2

Multiply both sides by

1 + \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}

\frac{1 - \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}}{1 + \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}} (1 + \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}) = 2 (1 + \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}})

Simplify

1 - \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 2 (1 + \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}})

Apply exponent rules

1 - \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 2 (1 + \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}})

Apply exponent rule:

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

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

1 - \frac{e ^{x} - ( e ^{x} ) ^{- 1}}{e ^{x} + ( e ^{x} ) ^{- 1}} = 2 (1 + \frac{e ^{x} - ( e ^{x} ) ^{- 1}}{e ^{x} + ( e ^{x} ) ^{- 1}})

1 - \frac{e ^{x} - ( e ^{x} ) ^{- 1}}{e ^{x} + ( e ^{x} ) ^{- 1}} = 2 (1 + \frac{e ^{x} - ( e ^{x} ) ^{- 1}}{e ^{x} + ( e ^{x} ) ^{- 1}})

Rewrite the equation with

e^{x} = u

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

Solve

1 - \frac{u - u ^{- 1}}{u + u ^{- 1}} = 2 (1 + \frac{u - u ^{- 1}}{u + u ^{- 1}}) : u = \frac{1}{2}, u = - \frac{1}{2}

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

Refine

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

Multiply both sides by

u^{2} + 1

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

Multiply both sides by

u^{2} + 1

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

Simplify

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

Simplify

1 \cdot (u^{2} + 1) : u^{2} + 1

1 \cdot (u^{2} + 1)

Multiply:

1 \cdot (u^{2} + 1) = (u^{2} + 1)

= (u^{2} + 1)

Remove parentheses:

(a) = a

= u^{2} + 1

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

u^{2} + 1

= - (u^{2} - 1)

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

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

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

Expand

u^{2} + 1 - (u^{2} - 1) : 2

u^{2} + 1 - (u^{2} - 1)

- (u^{2} - 1) : - u^{2} + 1

- (u^{2} - 1)

Distribute parentheses

= - (u^{2}) - (- 1)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - u^{2} + 1

= u^{2} + 1 - u^{2} + 1

Simplify

u^{2} + 1 - u^{2} + 1 : 2

u^{2} + 1 - u^{2} + 1

Group like terms

= u^{2} - u^{2} + 1 + 1

Add similar elements:

u^{2} - u^{2} = 0

= 1 + 1

Add the numbers:

1 + 1 = 2

= 2

= 2

Expand

2 (1 + \frac{u ^{2} - 1}{u ^{2} + 1}) (u^{2} + 1) : 4 u^{2}

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

Expand

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

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

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = 1, b = \frac{u ^{2} - 1}{u ^{2} + 1}, c = u^{2}, d = 1

= 1 \cdot u^{2} + 1 \cdot 1 + \frac{u ^{2} - 1}{u ^{2} + 1} u^{2} + \frac{u ^{2} - 1}{u ^{2} + 1} \cdot 1

= 1 \cdot u^{2} + 1 \cdot 1 + \frac{u ^{2} - 1}{u ^{2} + 1} u^{2} + 1 \cdot \frac{u ^{2} - 1}{u ^{2} + 1}

Simplify

1 \cdot u^{2} + 1 \cdot 1 + \frac{u ^{2} - 1}{u ^{2} + 1} u^{2} + 1 \cdot \frac{u ^{2} - 1}{u ^{2} + 1} : 2 u^{2}

1 \cdot u^{2} + 1 \cdot 1 + \frac{u ^{2} - 1}{u ^{2} + 1} u^{2} + 1 \cdot \frac{u ^{2} - 1}{u ^{2} + 1}

1 \cdot u^{2} = u^{2}

1 \cdot u^{2}

Multiply:

1 \cdot u^{2} = u^{2}

= u^{2}

1 \cdot 1 = 1

1 \cdot 1

Multiply the numbers:

1 \cdot 1 = 1

= 1

\frac{u ^{2} - 1}{u ^{2} + 1} u^{2} = \frac{u ^{4} - u ^{2}}{u ^{2} + 1}

\frac{u ^{2} - 1}{u ^{2} + 1} u^{2}

Multiply fractions:

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

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

Expand

(u^{2} - 1) u^{2} : u^{4} - u^{2}

(u^{2} - 1) u^{2}

= u^{2} (u^{2} - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = u^{2}, b = u^{2}, c = 1

= u^{2} u^{2} - u^{2} \cdot 1

= u^{2} u^{2} - 1 \cdot u^{2}

Simplify

u^{2} u^{2} - 1 \cdot u^{2} : u^{4} - u^{2}

u^{2} u^{2} - 1 \cdot u^{2}

u^{2} u^{2} = u^{4}

u^{2} u^{2}

Apply exponent rule:

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

u^{2} u^{2} = u^{2 + 2}

= u^{2 + 2}

Add the numbers:

2 + 2 = 4

= u^{4}

1 \cdot u^{2} = u^{2}

1 \cdot u^{2}

Multiply:

1 \cdot u^{2} = u^{2}

= u^{2}

= u^{4} - u^{2}

= u^{4} - u^{2}

= \frac{u ^{4} - u ^{2}}{u ^{2} + 1}

1 \cdot \frac{u ^{2} - 1}{u ^{2} + 1} = \frac{u ^{2} - 1}{u ^{2} + 1}

1 \cdot \frac{u ^{2} - 1}{u ^{2} + 1}

Multiply:

1 \cdot \frac{u ^{2} - 1}{u ^{2} + 1} = \frac{u ^{2} - 1}{u ^{2} + 1}

= \frac{u ^{2} - 1}{u ^{2} + 1}

= u^{2} + 1 + \frac{u ^{4} - u ^{2}}{u ^{2} + 1} + \frac{u ^{2} - 1}{u ^{2} + 1}

Combine the fractions

\frac{u ^{4} - u ^{2}}{u ^{2} + 1} + \frac{u ^{2} - 1}{u ^{2} + 1} : (u + 1) (u - 1)

Apply rule

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

= \frac{u ^{4} - u ^{2} + u ^{2} - 1}{u ^{2} + 1}

Add similar elements:

- u^{2} + u^{2} = 0

= \frac{u ^{4} - 1}{u ^{2} + 1}

Factor

u^{4} - 1 : (u^{2} + 1) (u + 1) (u - 1)

u^{4} - 1

Rewrite

u^{4} - 1

(u^{2})^{2} - 1^{2}

u^{4} - 1

Rewrite

1

1^{2}

= u^{4} - 1^{2}

Apply exponent rule:

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

u^{4} = (u^{2})^{2}

= (u^{2})^{2} - 1^{2}

= (u^{2})^{2} - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(u^{2})^{2} - 1^{2} = (u^{2} + 1) (u^{2} - 1)

= (u^{2} + 1) (u^{2} - 1)

Factor

u^{2} - 1 : (u + 1) (u - 1)

u^{2} - 1

Rewrite

1

1^{2}

= u^{2} - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

u^{2} - 1^{2} = (u + 1) (u - 1)

= (u + 1) (u - 1)

= (u^{2} + 1) (u + 1) (u - 1)

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

Cancel the common factor:

u^{2} + 1

= (u + 1) (u - 1)

= u^{2} + 1 + (u + 1) (u - 1)

Expand

(u + 1) (u - 1) : u^{2} - 1

(u + 1) (u - 1)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = u, b = 1

= u^{2} - 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= u^{2} - 1

= u^{2} + 1 + u^{2} - 1

Simplify

u^{2} + 1 + u^{2} - 1 : 2 u^{2}

u^{2} + 1 + u^{2} - 1

Group like terms

= u^{2} + u^{2} + 1 - 1

Add similar elements:

u^{2} + u^{2} = 2 u^{2}

= 2 u^{2} + 1 - 1

1 - 1 = 0

= 2 u^{2}

= 2 u^{2}

= 2 u^{2}

= 2 \cdot 2 u^{2}

Expand

2 \cdot 2 u^{2} : 4 u^{2}

2 \cdot 2 u^{2}

Distribute parentheses

= 2 \cdot 2 u^{2}

Multiply the numbers:

2 \cdot 2 = 4

= 4 u^{2}

= 4 u^{2}

2 = 4 u^{2}

Solve

2 = 4 u^{2} : u = \frac{1}{2}, u = - \frac{1}{2}

2 = 4 u^{2}

Switch sides

4 u^{2} = 2

Divide both sides by

4

4 u^{2} = 2

Divide both sides by

4

\frac{4 u ^{2}}{4} = \frac{2}{4}

Simplify

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

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

For

x^{2} = f (a)

the solutions are

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

u = \frac{1}{2}, u = - \frac{1}{2}

\frac{1}{2} = \frac{1}{2}

\frac{1}{2}

Apply radical rule:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= \frac{1}{2}

Apply radical rule:

1 = 1

1 = 1

= \frac{1}{2}

- \frac{1}{2} = - \frac{1}{2}

- \frac{1}{2}

Apply radical rule:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= - \frac{1}{2}

Apply radical rule:

1 = 1

1 = 1

= - \frac{1}{2}

u = \frac{1}{2}, u = - \frac{1}{2}

u = \frac{1}{2}, u = - \frac{1}{2}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

1 - \frac{u - u ^{- 1}}{u + u ^{- 1}}

and compare to zero

u = 0

Take the denominator(s) of

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

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{1}{2}, u = - \frac{1}{2}

u = \frac{1}{2}, u = - \frac{1}{2}

Substitute back

u = e^{x},

solve for

x

Solve

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

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

Apply exponent rules

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

Apply exponent rule:

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

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

e^{x} = 2^{- \frac{1}{2}}

Apply exponent rule:

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

e^{x} = 2^{- \frac{1}{2}}

f (x) = g (x)

, then

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

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

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

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

Apply log rule:

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

ln (2^{- \frac{1}{2}}) = - \frac{1}{2} ln (2)

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

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

Solve

e^{x} = - \frac{1}{2} :

No Solution for

x \in R

e^{x} = - \frac{1}{2}

Apply exponent rules

e^{x} = - \frac{1}{2}

Apply exponent rule:

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

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

e^{x} = - 2^{- \frac{1}{2}}

e^{x} = - 2^{- \frac{1}{2}}

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

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

Verify Solutions:

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

True

Check the solutions by plugging them into

\frac{1 - \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}}{1 + \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}} = 2

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

Plug in

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

True

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

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

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

Apply rule

- (- a) = a

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

\frac{e ^{- \frac{1}{2} l n (2)} - e ^{\frac{1}{2} l n (2)}}{e ^{- \frac{1}{2} l n (2)} + e ^{\frac{1}{2} l n (2)}} = - \frac{1}{3}

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

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

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

Apply exponent rule:

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

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

Apply log rule:

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

e^{l n (2)} = 2

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

Apply exponent rule:

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

= \frac{1}{2}

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

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

Apply exponent rule:

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

= e^{l n (2)}

Apply log rule:

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

e^{l n (2)} = 2

= 2

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

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

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

Apply exponent rule:

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

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

Apply log rule:

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

e^{l n (2)} = 2

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

Apply exponent rule:

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

= \frac{1}{2}

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

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

Apply exponent rule:

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

= e^{l n (2)}

Apply log rule:

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

e^{l n (2)} = 2

= 2

= \frac{\frac{1}{2} - 2}{\frac{1}{2} + 2}

Join

\frac{1}{2} + 2 : \frac{3}{2}

\frac{1}{2} + 2

Convert element to fraction:

2 = \frac{2 2}{2}

= \frac{1}{2} + \frac{2 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{1 + 2 2}{2}

1 + 22 = 3

1 + 22

Apply radical rule:

a a = a

22 = 2

= 1 + 2

Add the numbers:

1 + 2 = 3

= 3

= \frac{3}{2}

= \frac{\frac{1}{2} - 2}{\frac{3}{2}}

Join

\frac{1}{2} - 2 : - \frac{1}{2}

\frac{1}{2} - 2

Convert element to fraction:

2 = \frac{2 2}{2}

= \frac{1}{2} - \frac{2 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{1 - 2 2}{2}

1 - 22 = - 1

1 - 22

Apply radical rule:

a a = a

22 = 2

= 1 - 2

Subtract the numbers:

1 - 2 = - 1

= - 1

= \frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

= \frac{- \frac{1}{2}}{\frac{3}{2}}

Apply the fraction rule:

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

= - \frac{\frac{1}{2}}{\frac{3}{2}}

Divide fractions:

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

= - \frac{1 \cdot 2}{2 \cdot 3}

Refine

= - \frac{2}{2 \cdot 3}

Cancel the common factor:

2

= - \frac{1}{3}

= \frac{1 - \frac{e ^{- \frac{1}{2} l n (2)} - e ^{\frac{1}{2} l n (2)}}{e ^{- \frac{1}{2} l n (2)} + e ^{\frac{1}{2} l n (2)}}}{1 - \frac{1}{3}}

\frac{e ^{- \frac{1}{2} l n (2)} - e ^{\frac{1}{2} l n (2)}}{e ^{- \frac{1}{2} l n (2)} + e ^{\frac{1}{2} l n (2)}} = - \frac{1}{3}

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

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

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

Apply exponent rule:

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

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

Apply log rule:

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

e^{l n (2)} = 2

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

Apply exponent rule:

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

= \frac{1}{2}

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

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

Apply exponent rule:

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

= e^{l n (2)}

Apply log rule:

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

e^{l n (2)} = 2

= 2

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

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

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

Apply exponent rule:

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

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

Apply log rule:

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

e^{l n (2)} = 2

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

Apply exponent rule:

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

= \frac{1}{2}

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

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

Apply exponent rule:

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

= e^{l n (2)}

Apply log rule:

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

e^{l n (2)} = 2

= 2

= \frac{\frac{1}{2} - 2}{\frac{1}{2} + 2}

Join

\frac{1}{2} + 2 : \frac{3}{2}

\frac{1}{2} + 2

Convert element to fraction:

2 = \frac{2 2}{2}

= \frac{1}{2} + \frac{2 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{1 + 2 2}{2}

1 + 22 = 3

1 + 22

Apply radical rule:

a a = a

22 = 2

= 1 + 2

Add the numbers:

1 + 2 = 3

= 3

= \frac{3}{2}

= \frac{\frac{1}{2} - 2}{\frac{3}{2}}

Join

\frac{1}{2} - 2 : - \frac{1}{2}

\frac{1}{2} - 2

Convert element to fraction:

2 = \frac{2 2}{2}

= \frac{1}{2} - \frac{2 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{1 - 2 2}{2}

1 - 22 = - 1

1 - 22

Apply radical rule:

a a = a

22 = 2

= 1 - 2

Subtract the numbers:

1 - 2 = - 1

= - 1

= \frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

= \frac{- \frac{1}{2}}{\frac{3}{2}}

Apply the fraction rule:

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

= - \frac{\frac{1}{2}}{\frac{3}{2}}

Divide fractions:

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

= - \frac{1 \cdot 2}{2 \cdot 3}

Refine

= - \frac{2}{2 \cdot 3}

Cancel the common factor:

2

= - \frac{1}{3}

= \frac{1 - ( - \frac{1}{3} )}{1 - \frac{1}{3}}

Simplify

\frac{1 - ( - \frac{1}{3} )}{1 - \frac{1}{3}}

Apply rule

- (- a) = a

= \frac{1 + \frac{1}{3}}{1 - \frac{1}{3}}

Join

1 - \frac{1}{3} : \frac{2}{3}

1 - \frac{1}{3}

Convert element to fraction:

1 = \frac{1 \cdot 3}{3}

= \frac{1 \cdot 3}{3} - \frac{1}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 3 - 1}{3}

1 \cdot 3 - 1 = 2

1 \cdot 3 - 1

Multiply the numbers:

1 \cdot 3 = 3

= 3 - 1

Subtract the numbers:

3 - 1 = 2

= 2

= \frac{2}{3}

= \frac{1 + \frac{1}{3}}{\frac{2}{3}}

Join

1 + \frac{1}{3} : \frac{4}{3}

1 + \frac{1}{3}

Convert element to fraction:

1 = \frac{1 \cdot 3}{3}

= \frac{1 \cdot 3}{3} + \frac{1}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 3 + 1}{3}

1 \cdot 3 + 1 = 4

1 \cdot 3 + 1

Multiply the numbers:

1 \cdot 3 = 3

= 3 + 1

Add the numbers:

3 + 1 = 4

= 4

= \frac{4}{3}

= \frac{\frac{4}{3}}{\frac{2}{3}}

Divide fractions:

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

= \frac{4 \cdot 3}{3 \cdot 2}

Cancel the common factor:

3

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

= 2

2 = 2

True

The solution is

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

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

Popular Examples

40=tan(ln(18+1)+C)

40 = tan (ln (18 + 1) + C)

tan(a)=-2sqrt(6)

tan (a) = - 26

-5cos(x)=2sin^2(x)+4

- 5 cos (x) = 2 sin^{2} (x) + 4

sqrt(5)sin(θ+0.4636)=1

5 sin (θ + 0.4636) = 1

0= 20/3 sin(10t)+5cos(10t)

0 = \frac{20}{3} sin (10 t) + 5 cos (10 t)

Frequently Asked Questions (FAQ)

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