What is the value of coth(ln(4)) ?

The value of coth(ln(4)) is 17/15

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coth(ln(4))

Solution

coth (ln (4))

Solution

\frac{17}{15}

Decimal Notation

1.13333 \dots

Solution steps

coth (ln (4))

Simplify:

ln (4) = 2 ln (2)

ln (4)

Rewrite

4

in power-base form:

4 = 2^{2}

= ln (2^{2})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (2^{2}) = 2 ln (2)

= 2 ln (2)

= coth (2 ln (2))

Rewrite using trig identities:

\frac{17}{15}

coth (2 ln (2))

Use the Hyperbolic identity:

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

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

\frac{e ^{2 l n (2)} + e ^{- 2 l n (2)}}{e ^{2 l n (2)} - e ^{- 2 l n (2)}} = \frac{17}{15}

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

e^{2 l n (2)} = 2^{2}

e^{2 l n (2)}

Apply exponent rule:

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

= (e^{l n (2)})^{2}

Apply log rule:

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

e^{l n (2)} = 2

= 2^{2}

e^{- 2 l n (2)} = 2^{- 2}

e^{- 2 l n (2)}

Apply exponent rule:

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

= (e^{l n (2)})^{- 2}

Apply log rule:

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

e^{l n (2)} = 2

= 2^{- 2}

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

e^{2 l n (2)} = 2^{2}

e^{2 l n (2)}

Apply exponent rule:

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

= (e^{l n (2)})^{2}

Apply log rule:

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

e^{l n (2)} = 2

= 2^{2}

e^{- 2 l n (2)} = 2^{- 2}

e^{- 2 l n (2)}

Apply exponent rule:

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

= (e^{l n (2)})^{- 2}

Apply log rule:

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

e^{l n (2)} = 2

= 2^{- 2}

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

Simplify

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

2^{2} = 4

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

2^{2} = 4

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

Apply exponent rule:

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

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

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

Apply exponent rule:

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

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

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

2^{2} = 4

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

2^{2} = 4

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

Join

4 - \frac{1}{4} : \frac{15}{4}

4 - \frac{1}{4}

Convert element to fraction:

4 = \frac{4 \cdot 4}{4}

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

Since the denominators are equal, combine the fractions:

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

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

4 \cdot 4 - 1 = 15

4 \cdot 4 - 1

Multiply the numbers:

4 \cdot 4 = 16

= 16 - 1

Subtract the numbers:

16 - 1 = 15

= 15

= \frac{15}{4}

= \frac{4 + \frac{1}{4}}{\frac{15}{4}}

Join

4 + \frac{1}{4} : \frac{17}{4}

4 + \frac{1}{4}

Convert element to fraction:

4 = \frac{4 \cdot 4}{4}

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

Since the denominators are equal, combine the fractions:

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

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

4 \cdot 4 + 1 = 17

4 \cdot 4 + 1

Multiply the numbers:

4 \cdot 4 = 16

= 16 + 1

Add the numbers:

16 + 1 = 17

= 17

= \frac{17}{4}

= \frac{\frac{17}{4}}{\frac{15}{4}}

Divide fractions:

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

= \frac{17 \cdot 4}{4 \cdot 15}

Cancel the common factor:

4

= \frac{17}{15}

= \frac{17}{15}

= \frac{17}{15}

= \frac{17}{15}

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Frequently Asked Questions (FAQ)

What is the value of coth(ln(4)) ?
The value of coth(ln(4)) is 17/15