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

The general solution for ((1-tanh(2x)))/((1+tanh(2x)))=2 is x=-1/4 ln(2)

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

Solution

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

Solution

x = - \frac{1}{4} ln (2)

Degrees

x = - 9.92860 \dots^{\circ}

Solution steps

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

Rewrite using trig identities

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

Use the Hyperbolic identity:

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

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

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

\frac{1 - \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}}}{1 + \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}}} = 2 : x = - \frac{1}{4} ln (2)

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

Multiply both sides by

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

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

Simplify

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

Apply exponent rules

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

Apply exponent rule:

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

e^{2 x} = (e^{x})^{2}, e^{- 2 x} = (e^{x})^{- 2}

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

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

Rewrite the equation with

e^{x} = u

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

Solve

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

Refine

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

Multiply both sides by

u^{4} + 1

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

Multiply both sides by

u^{4} + 1

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

Simplify

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

Simplify

1 \cdot (u^{4} + 1) : u^{4} + 1

1 \cdot (u^{4} + 1)

Multiply:

1 \cdot (u^{4} + 1) = (u^{4} + 1)

= (u^{4} + 1)

Remove parentheses:

(a) = a

= u^{4} + 1

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

u^{4} + 1

= - (u^{4} - 1)

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

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

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

Expand

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

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

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

- (u^{4} - 1)

Distribute parentheses

= - (u^{4}) - (- 1)

Apply minus-plus rules

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

= - u^{4} + 1

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

Simplify

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

u^{4} + 1 - u^{4} + 1

Group like terms

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

Add similar elements:

u^{4} - u^{4} = 0

= 1 + 1

Add the numbers:

1 + 1 = 2

= 2

= 2

Expand

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

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

Expand

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

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

Apply FOIL method:

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

a = 1, b = \frac{u ^{4} - 1}{u ^{4} + 1}, c = u^{4}, d = 1

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

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

Simplify

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

1 \cdot u^{4} + 1 \cdot 1 + \frac{u ^{4} - 1}{u ^{4} + 1} u^{4} + 1 \cdot \frac{u ^{4} - 1}{u ^{4} + 1}

1 \cdot u^{4} = u^{4}

1 \cdot u^{4}

Multiply:

1 \cdot u^{4} = u^{4}

= u^{4}

1 \cdot 1 = 1

1 \cdot 1

Multiply the numbers:

1 \cdot 1 = 1

= 1

\frac{u ^{4} - 1}{u ^{4} + 1} u^{4} = \frac{u ^{8} - u ^{4}}{u ^{4} + 1}

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

Multiply fractions:

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

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

Expand

(u^{4} - 1) u^{4} : u^{8} - u^{4}

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

= u^{4} (u^{4} - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = u^{4}, b = u^{4}, c = 1

= u^{4} u^{4} - u^{4} \cdot 1

= u^{4} u^{4} - 1 \cdot u^{4}

Simplify

u^{4} u^{4} - 1 \cdot u^{4} : u^{8} - u^{4}

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

u^{4} u^{4} = u^{8}

u^{4} u^{4}

Apply exponent rule:

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

u^{4} u^{4} = u^{4 + 4}

= u^{4 + 4}

Add the numbers:

4 + 4 = 8

= u^{8}

1 \cdot u^{4} = u^{4}

1 \cdot u^{4}

Multiply:

1 \cdot u^{4} = u^{4}

= u^{4}

= u^{8} - u^{4}

= u^{8} - u^{4}

= \frac{u ^{8} - u ^{4}}{u ^{4} + 1}

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

1 \cdot \frac{u ^{4} - 1}{u ^{4} + 1}

Multiply:

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

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

= u^{4} + 1 + \frac{u ^{8} - u ^{4}}{u ^{4} + 1} + \frac{u ^{4} - 1}{u ^{4} + 1}

Combine the fractions

\frac{u ^{8} - u ^{4}}{u ^{4} + 1} + \frac{u ^{4} - 1}{u ^{4} + 1} : (u^{2} + 1) (u + 1) (u - 1)

Apply rule

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

= \frac{u ^{8} - u ^{4} + u ^{4} - 1}{u ^{4} + 1}

Add similar elements:

- u^{4} + u^{4} = 0

= \frac{u ^{8} - 1}{u ^{4} + 1}

Factor

u^{8} - 1 : (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) (u^{2} + 1) (u + 1) (u - 1)

u^{8} - 1

Rewrite

u^{8} - 1

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

u^{8} - 1

Rewrite

1

1^{2}

= u^{8} - 1^{2}

Apply exponent rule:

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

u^{8} = (u^{4})^{2}

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

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

Apply Difference of Two Squares Formula:

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

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

= (u^{4} + 1) (u^{4} - 1)

Factor

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

u^{4} + 1

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

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

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

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

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

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

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

Cancel

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

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

Cancel the common factor:

u^{2} + 2 u + 1

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

Cancel the common factor:

u^{2} - 2 u + 1

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

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

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

Expand

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

Expand

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

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

Apply Difference of Two Squares Formula:

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

a = u^{2}, b = 1

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

Simplify

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

(u^{2})^{2}

Apply exponent rule:

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

= u^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= u^{4}

= u^{4} - 1

= u^{4} - 1

= u^{4} - 1

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

Simplify

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

u^{4} + 1 + u^{4} - 1

Group like terms

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

Add similar elements:

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

= 2 u^{4} + 1 - 1

1 - 1 = 0

= 2 u^{4}

= 2 u^{4}

= 2 u^{4}

= 2 \cdot 2 u^{4}

Expand

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

2 \cdot 2 u^{4}

Distribute parentheses

= 2 \cdot 2 u^{4}

Multiply the numbers:

2 \cdot 2 = 4

= 4 u^{4}

= 4 u^{4}

2 = 4 u^{4}

Solve

2 = 4 u^{4}

Switch sides

4 u^{4} = 2

Divide both sides by

4

4 u^{4} = 2

Divide both sides by

4

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

Simplify

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

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

For

x^{n} = f (a)

, n is even, the solutions are

Apply radical rule:

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

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

and compare to zero

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

u = 0

Take the denominator(s) of

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

and compare to zero

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

Substitute back

u = e^{x},

solve for

x

Solve

Apply exponent rules

Apply exponent rule:

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

e^{x} = 2^{- \frac{1}{4}}

Apply exponent rule:

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

e^{x} = 2^{- \frac{1}{4}}

f (x) = g (x)

, then

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

ln (e^{x}) = ln (2^{- \frac{1}{4}})

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (2^{- \frac{1}{4}})

Apply log rule:

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

ln (2^{- \frac{1}{4}}) = - \frac{1}{4} ln (2)

x = - \frac{1}{4} ln (2)

x = - \frac{1}{4} ln (2)

Solve

No Solution for

x \in R

Apply exponent rules

Apply exponent rule:

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

e^{x} = - 2^{- \frac{1}{4}}

e^{x} = - 2^{- \frac{1}{4}}

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = - \frac{1}{4} ln (2)

Verify Solutions:

x = - \frac{1}{4} ln (2)

True

Check the solutions by plugging them into

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

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

Plug in

x = - \frac{1}{4} ln (2) :

True

\frac{1 - \frac{e ^{2 (- \frac{1}{4} l n (2))} - e ^{- 2 (- \frac{1}{4} l n (2))}}{e ^{2 (- \frac{1}{4} l n (2))} + e ^{- 2 (- \frac{1}{4} l n (2))}}}{1 + \frac{e ^{2 (- \frac{1}{4} l n (2))} - e ^{- 2 (- \frac{1}{4} l n (2))}}{e ^{2 (- \frac{1}{4} l n (2))} + e ^{- 2 (- \frac{1}{4} l n (2))}}} = 2

\frac{1 - \frac{e ^{2 (- \frac{1}{4} l n (2))} - e ^{- 2 (- \frac{1}{4} l n (2))}}{e ^{2 (- \frac{1}{4} l n (2))} + e ^{- 2 (- \frac{1}{4} l n (2))}}}{1 + \frac{e ^{2 (- \frac{1}{4} l n (2))} - e ^{- 2 (- \frac{1}{4} l n (2))}}{e ^{2 (- \frac{1}{4} l n (2))} + e ^{- 2 (- \frac{1}{4} l n (2))}}} = 2

\frac{1 - \frac{e ^{2 (- \frac{1}{4} l n (2))} - e ^{- 2 (- \frac{1}{4} l n (2))}}{e ^{2 (- \frac{1}{4} l n (2))} + e ^{- 2 (- \frac{1}{4} l n (2))}}}{1 + \frac{e ^{2 (- \frac{1}{4} l n (2))} - e ^{- 2 (- \frac{1}{4} l n (2))}}{e ^{2 (- \frac{1}{4} l n (2))} + e ^{- 2 (- \frac{1}{4} l n (2))}}}

Remove parentheses:

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

= \frac{1 - \frac{e ^{- 2 \cdot \frac{1}{4} l n (2)} - e ^{2 \cdot \frac{1}{4} l n (2)}}{e ^{- 2 \cdot \frac{1}{4} l n (2)} + e ^{2 \cdot \frac{1}{4} l n (2)}}}{1 + \frac{e ^{- 2 \cdot \frac{1}{4} l n (2)} - e ^{2 \cdot \frac{1}{4} l n (2)}}{e ^{- 2 \cdot \frac{1}{4} l n (2)} + e ^{2 \cdot \frac{1}{4} l n (2)}}}

\frac{e ^{- 2 \cdot \frac{1}{4} l n (2)} - e ^{2 \cdot \frac{1}{4} l n (2)}}{e ^{- 2 \cdot \frac{1}{4} l n (2)} + e ^{2 \cdot \frac{1}{4} l n (2)}} = - \frac{1}{3}

\frac{e ^{- 2 \cdot \frac{1}{4} l n (2)} - e ^{2 \cdot \frac{1}{4} l n (2)}}{e ^{- 2 \cdot \frac{1}{4} l n (2)} + e ^{2 \cdot \frac{1}{4} l n (2)}}

e^{- 2 \cdot \frac{1}{4} l n (2)} = \frac{1}{2}

e^{- 2 \cdot \frac{1}{4} l n (2)}

Multiply

- 2 \cdot \frac{1}{4} ln (2) : - \frac{1}{2} ln (2)

- 2 \cdot \frac{1}{4} ln (2)

Multiply fractions:

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

= - \frac{1 \cdot 2}{4} ln (2)

\frac{1 \cdot 2}{4} = \frac{1}{2}

\frac{1 \cdot 2}{4}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{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^{2 \cdot \frac{1}{4} l n (2)} = 2

e^{2 \cdot \frac{1}{4} l n (2)}

Multiply

2 \cdot \frac{1}{4} ln (2) : \frac{1}{2} ln (2)

2 \cdot \frac{1}{4} ln (2)

Multiply fractions:

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

= \frac{1 \cdot 2}{4} ln (2)

\frac{1 \cdot 2}{4} = \frac{1}{2}

\frac{1 \cdot 2}{4}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

= \frac{1}{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 ^{- 2 \cdot \frac{1}{4} l n (2)} - e ^{2 \cdot \frac{1}{4} l n (2)}}{\frac{1}{2} + 2}

e^{- 2 \cdot \frac{1}{4} l n (2)} = \frac{1}{2}

e^{- 2 \cdot \frac{1}{4} l n (2)}

Multiply

- 2 \cdot \frac{1}{4} ln (2) : - \frac{1}{2} ln (2)

- 2 \cdot \frac{1}{4} ln (2)

Multiply fractions:

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

= - \frac{1 \cdot 2}{4} ln (2)

\frac{1 \cdot 2}{4} = \frac{1}{2}

\frac{1 \cdot 2}{4}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{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^{2 \cdot \frac{1}{4} l n (2)} = 2

e^{2 \cdot \frac{1}{4} l n (2)}

Multiply

2 \cdot \frac{1}{4} ln (2) : \frac{1}{2} ln (2)

2 \cdot \frac{1}{4} ln (2)

Multiply fractions:

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

= \frac{1 \cdot 2}{4} ln (2)

\frac{1 \cdot 2}{4} = \frac{1}{2}

\frac{1 \cdot 2}{4}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

= \frac{1}{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 ^{- 2 \cdot \frac{1}{4} l n (2)} - e ^{2 \cdot \frac{1}{4} l n (2)}}{e ^{- 2 \cdot \frac{1}{4} l n (2)} + e ^{2 \cdot \frac{1}{4} l n (2)}}}{1 - \frac{1}{3}}

\frac{e ^{- 2 \cdot \frac{1}{4} l n (2)} - e ^{2 \cdot \frac{1}{4} l n (2)}}{e ^{- 2 \cdot \frac{1}{4} l n (2)} + e ^{2 \cdot \frac{1}{4} l n (2)}} = - \frac{1}{3}

\frac{e ^{- 2 \cdot \frac{1}{4} l n (2)} - e ^{2 \cdot \frac{1}{4} l n (2)}}{e ^{- 2 \cdot \frac{1}{4} l n (2)} + e ^{2 \cdot \frac{1}{4} l n (2)}}

e^{- 2 \cdot \frac{1}{4} l n (2)} = \frac{1}{2}

e^{- 2 \cdot \frac{1}{4} l n (2)}

Multiply

- 2 \cdot \frac{1}{4} ln (2) : - \frac{1}{2} ln (2)

- 2 \cdot \frac{1}{4} ln (2)

Multiply fractions:

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

= - \frac{1 \cdot 2}{4} ln (2)

\frac{1 \cdot 2}{4} = \frac{1}{2}

\frac{1 \cdot 2}{4}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{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^{2 \cdot \frac{1}{4} l n (2)} = 2

e^{2 \cdot \frac{1}{4} l n (2)}

Multiply

2 \cdot \frac{1}{4} ln (2) : \frac{1}{2} ln (2)

2 \cdot \frac{1}{4} ln (2)

Multiply fractions:

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

= \frac{1 \cdot 2}{4} ln (2)

\frac{1 \cdot 2}{4} = \frac{1}{2}

\frac{1 \cdot 2}{4}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

= \frac{1}{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 ^{- 2 \cdot \frac{1}{4} l n (2)} - e ^{2 \cdot \frac{1}{4} l n (2)}}{\frac{1}{2} + 2}

e^{- 2 \cdot \frac{1}{4} l n (2)} = \frac{1}{2}

e^{- 2 \cdot \frac{1}{4} l n (2)}

Multiply

- 2 \cdot \frac{1}{4} ln (2) : - \frac{1}{2} ln (2)

- 2 \cdot \frac{1}{4} ln (2)

Multiply fractions:

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

= - \frac{1 \cdot 2}{4} ln (2)

\frac{1 \cdot 2}{4} = \frac{1}{2}

\frac{1 \cdot 2}{4}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{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^{2 \cdot \frac{1}{4} l n (2)} = 2

e^{2 \cdot \frac{1}{4} l n (2)}

Multiply

2 \cdot \frac{1}{4} ln (2) : \frac{1}{2} ln (2)

2 \cdot \frac{1}{4} ln (2)

Multiply fractions:

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

= \frac{1 \cdot 2}{4} ln (2)

\frac{1 \cdot 2}{4} = \frac{1}{2}

\frac{1 \cdot 2}{4}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

= \frac{1}{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}{4} ln (2)

x = - \frac{1}{4} ln (2)

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Frequently Asked Questions (FAQ)

What is the general solution for ((1-tanh(2x)))/((1+tanh(2x)))=2 ?
The general solution for ((1-tanh(2x)))/((1+tanh(2x)))=2 is x=-1/4 ln(2)