What is the general solution for tanh(x)= 12/13 ?

The general solution for tanh(x)= 12/13 is x=ln(5)

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tanh(x)= 12/13

Solution

tanh (x) = \frac{12}{13}

Solution

x = ln (5)

Degrees

x = 92.21399 \dots^{\circ}

Solution steps

tanh (x) = \frac{12}{13}

Rewrite using trig identities

tanh (x) = \frac{12}{13}

Use the Hyperbolic identity:

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

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = \frac{12}{13}

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = \frac{12}{13}

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = \frac{12}{13} : x = ln (5)

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = \frac{12}{13}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

(e^{x} - e^{- x}) \cdot 13 = (e^{x} + e^{- x}) \cdot 12

Apply exponent rules

(e^{x} - e^{- x}) \cdot 13 = (e^{x} + e^{- x}) \cdot 12

Apply exponent rule:

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

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

(e^{x} - (e^{x})^{- 1}) \cdot 13 = (e^{x} + (e^{x})^{- 1}) \cdot 12

(e^{x} - (e^{x})^{- 1}) \cdot 13 = (e^{x} + (e^{x})^{- 1}) \cdot 12

Rewrite the equation with

e^{x} = u

(u - (u)^{- 1}) \cdot 13 = (u + (u)^{- 1}) \cdot 12

Solve

(u - u^{- 1}) \cdot 13 = (u + u^{- 1}) \cdot 12 : u = 5, u = - 5

(u - u^{- 1}) \cdot 13 = (u + u^{- 1}) \cdot 12

Refine

(u - \frac{1}{u}) \cdot 13 = (u + \frac{1}{u}) \cdot 12

Simplify

(u - \frac{1}{u}) \cdot 13 = (u + \frac{1}{u}) \cdot 12

Simplify

(u - \frac{1}{u}) \cdot 13 : 13 (u - \frac{1}{u})

(u - \frac{1}{u}) \cdot 13

Apply the commutative law:

(u - \frac{1}{u}) \cdot 13 = 13 (u - \frac{1}{u})

13 (u - \frac{1}{u})

Simplify

(u + \frac{1}{u}) \cdot 12 : 12 (u + \frac{1}{u})

(u + \frac{1}{u}) \cdot 12

Apply the commutative law:

(u + \frac{1}{u}) \cdot 12 = 12 (u + \frac{1}{u})

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

13 (u - \frac{1}{u}) = 12 (u + \frac{1}{u})

13 (u - \frac{1}{u}) = 12 (u + \frac{1}{u})

Expand

13 (u - \frac{1}{u}) : 13 u - \frac{13}{u}

13 (u - \frac{1}{u})

Apply the distributive law:

a (b - c) = ab - a c

a = 13, b = u, c = \frac{1}{u}

= 13 u - 13 \cdot \frac{1}{u}

13 \cdot \frac{1}{u} = \frac{13}{u}

13 \cdot \frac{1}{u}

Multiply fractions:

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

= \frac{1 \cdot 13}{u}

Multiply the numbers:

1 \cdot 13 = 13

= \frac{13}{u}

= 13 u - \frac{13}{u}

Expand

12 (u + \frac{1}{u}) : 12 u + \frac{12}{u}

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

Apply the distributive law:

a (b + c) = ab + a c

a = 12, b = u, c = \frac{1}{u}

= 12 u + 12 \cdot \frac{1}{u}

12 \cdot \frac{1}{u} = \frac{12}{u}

12 \cdot \frac{1}{u}

Multiply fractions:

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

= \frac{1 \cdot 12}{u}

Multiply the numbers:

1 \cdot 12 = 12

= \frac{12}{u}

= 12 u + \frac{12}{u}

13 u - \frac{13}{u} = 12 u + \frac{12}{u}

Multiply both sides by

u

13 u - \frac{13}{u} = 12 u + \frac{12}{u}

Multiply both sides by

u

13 uu - \frac{13}{u} u = 12 uu + \frac{12}{u} u

Simplify

13 uu - \frac{13}{u} u = 12 uu + \frac{12}{u} u

Simplify

13 uu : 13 u^{2}

13 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 13 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 13 u^{2}

Simplify

- \frac{13}{u} u : - 13

- \frac{13}{u} u

Multiply fractions:

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

= - \frac{13 u}{u}

Cancel the common factor:

u

= - 13

Simplify

12 uu : 12 u^{2}

12 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 12 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 12 u^{2}

Simplify

\frac{12}{u} u : 12

\frac{12}{u} u

Multiply fractions:

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

= \frac{12 u}{u}

Cancel the common factor:

u

= 12

13 u^{2} - 13 = 12 u^{2} + 12

13 u^{2} - 13 = 12 u^{2} + 12

13 u^{2} - 13 = 12 u^{2} + 12

Solve

13 u^{2} - 13 = 12 u^{2} + 12 : u = 5, u = - 5

13 u^{2} - 13 = 12 u^{2} + 12

Move

13

to the right side

13 u^{2} - 13 = 12 u^{2} + 12

Add

13

to both sides

13 u^{2} - 13 + 13 = 12 u^{2} + 12 + 13

Simplify

13 u^{2} = 12 u^{2} + 25

13 u^{2} = 12 u^{2} + 25

Move

12 u^{2}

to the left side

13 u^{2} = 12 u^{2} + 25

Subtract

12 u^{2}

from both sides

13 u^{2} - 12 u^{2} = 12 u^{2} + 25 - 12 u^{2}

Simplify

u^{2} = 25

u^{2} = 25

For

x^{2} = f (a)

the solutions are

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

u = 25, u = - 25

25 = 5

25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

a^{2} = a, a \geq 0

5^{2} = 5

= 5

- 25 = - 5

- 25

Factor the number:

25 = 5^{2}

= - 5^{2}

Apply radical rule:

a^{2} = a, a \geq 0

5^{2} = - 5

= - 5

u = 5, u = - 5

u = 5, u = - 5

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

(u - u^{- 1}) 13

and compare to zero

u = 0

Take the denominator(s) of

(u + u^{- 1}) 12

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 5, u = - 5

u = 5, u = - 5

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = 5 : x = ln (5)

e^{x} = 5

Apply exponent rules

e^{x} = 5

f (x) = g (x)

, then

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

ln (e^{x}) = ln (5)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (5)

x = ln (5)

Solve

e^{x} = - 5 :

No Solution for

x \in R

e^{x} = - 5

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = ln (5)

Verify Solutions:

x = ln (5)

True

Check the solutions by plugging them into

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = \frac{12}{13}

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

Plug in

x = ln (5) :

True

\frac{e ^{l n (5)} - e ^{- l n (5)}}{e ^{l n (5)} + e ^{- l n (5)}} = \frac{12}{13}

\frac{e ^{l n (5)} - e ^{- l n (5)}}{e ^{l n (5)} + e ^{- l n (5)}} = \frac{12}{13}

\frac{e ^{l n (5)} - e ^{- l n (5)}}{e ^{l n (5)} + e ^{- l n (5)}}

e^{l n (5)} = 5

e^{l n (5)}

Apply log rule:

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

= 5

e^{- l n (5)} = 5^{- 1}

e^{- l n (5)}

Apply exponent rule:

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

= (e^{l n (5)})^{- 1}

Apply log rule:

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

e^{l n (5)} = 5

= 5^{- 1}

= \frac{e ^{l n (5)} - e ^{- l n (5)}}{5 + 5 ^{- 1}}

e^{l n (5)} = 5

e^{l n (5)}

Apply log rule:

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

= 5

e^{- l n (5)} = 5^{- 1}

e^{- l n (5)}

Apply exponent rule:

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

= (e^{l n (5)})^{- 1}

Apply log rule:

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

e^{l n (5)} = 5

= 5^{- 1}

= \frac{5 - 5 ^{- 1}}{5 + 5 ^{- 1}}

Simplify

\frac{5 - 5 ^{- 1}}{5 + 5 ^{- 1}}

Apply exponent rule:

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

5^{- 1} = \frac{1}{5}

= \frac{5 - 5 ^{- 1}}{5 + \frac{1}{5}}

Apply exponent rule:

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

5^{- 1} = \frac{1}{5}

= \frac{5 - \frac{1}{5}}{5 + \frac{1}{5}}

Join

5 + \frac{1}{5} : \frac{26}{5}

5 + \frac{1}{5}

Convert element to fraction:

5 = \frac{5 \cdot 5}{5}

= \frac{5 \cdot 5}{5} + \frac{1}{5}

Since the denominators are equal, combine the fractions:

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

= \frac{5 \cdot 5 + 1}{5}

5 \cdot 5 + 1 = 26

5 \cdot 5 + 1

Multiply the numbers:

5 \cdot 5 = 25

= 25 + 1

Add the numbers:

25 + 1 = 26

= 26

= \frac{26}{5}

= \frac{5 - \frac{1}{5}}{\frac{26}{5}}

Join

5 - \frac{1}{5} : \frac{24}{5}

5 - \frac{1}{5}

Convert element to fraction:

5 = \frac{5 \cdot 5}{5}

= \frac{5 \cdot 5}{5} - \frac{1}{5}

Since the denominators are equal, combine the fractions:

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

= \frac{5 \cdot 5 - 1}{5}

5 \cdot 5 - 1 = 24

5 \cdot 5 - 1

Multiply the numbers:

5 \cdot 5 = 25

= 25 - 1

Subtract the numbers:

25 - 1 = 24

= 24

= \frac{24}{5}

= \frac{\frac{24}{5}}{\frac{26}{5}}

Divide fractions:

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

= \frac{24 \cdot 5}{5 \cdot 26}

Cancel the common factor:

5

= \frac{24}{26}

Cancel the common factor:

2

= \frac{12}{13}

= \frac{12}{13}

\frac{12}{13} = \frac{12}{13}

True

The solution is

x = ln (5)

x = ln (5)

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Frequently Asked Questions (FAQ)

What is the general solution for tanh(x)= 12/13 ?
The general solution for tanh(x)= 12/13 is x=ln(5)