What is the value of coth(ln(7)) ?

The value of coth(ln(7)) is 25/24

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Popular >

coth(ln(7))

Solution

coth (ln (7))

Solution

\frac{25}{24}

Decimal Notation

1.04166 \dots

Solution steps

coth (ln (7))

Use the Hyperbolic identity:

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

= \frac{e ^{l n (7)} + e ^{- l n (7)}}{e ^{l n (7)} - e ^{- l n (7)}}

\frac{e ^{l n (7)} + e ^{- l n (7)}}{e ^{l n (7)} - e ^{- l n (7)}} = \frac{25}{24}

\frac{e ^{l n (7)} + e ^{- l n (7)}}{e ^{l n (7)} - e ^{- l n (7)}}

e^{l n (7)} = 7

e^{l n (7)}

Apply log rule:

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

= 7

e^{- l n (7)} = 7^{- 1}

e^{- l n (7)}

Apply exponent rule:

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

= (e^{l n (7)})^{- 1}

Apply log rule:

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

e^{l n (7)} = 7

= 7^{- 1}

= \frac{e ^{l n (7)} + e ^{- l n (7)}}{7 - 7 ^{- 1}}

e^{l n (7)} = 7

e^{l n (7)}

Apply log rule:

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

= 7

e^{- l n (7)} = 7^{- 1}

e^{- l n (7)}

Apply exponent rule:

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

= (e^{l n (7)})^{- 1}

Apply log rule:

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

e^{l n (7)} = 7

= 7^{- 1}

= \frac{7 + 7 ^{- 1}}{7 - 7 ^{- 1}}

Simplify

\frac{7 + 7 ^{- 1}}{7 - 7 ^{- 1}}

Apply exponent rule:

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

7^{- 1} = \frac{1}{7}

= \frac{7 + 7 ^{- 1}}{7 - \frac{1}{7}}

Apply exponent rule:

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

7^{- 1} = \frac{1}{7}

= \frac{7 + \frac{1}{7}}{7 - \frac{1}{7}}

Join

7 - \frac{1}{7} : \frac{48}{7}

7 - \frac{1}{7}

Convert element to fraction:

7 = \frac{7 \cdot 7}{7}

= \frac{7 \cdot 7}{7} - \frac{1}{7}

Since the denominators are equal, combine the fractions:

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

= \frac{7 \cdot 7 - 1}{7}

7 \cdot 7 - 1 = 48

7 \cdot 7 - 1

Multiply the numbers:

7 \cdot 7 = 49

= 49 - 1

Subtract the numbers:

49 - 1 = 48

= 48

= \frac{48}{7}

= \frac{7 + \frac{1}{7}}{\frac{48}{7}}

Join

7 + \frac{1}{7} : \frac{50}{7}

7 + \frac{1}{7}

Convert element to fraction:

7 = \frac{7 \cdot 7}{7}

= \frac{7 \cdot 7}{7} + \frac{1}{7}

Since the denominators are equal, combine the fractions:

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

= \frac{7 \cdot 7 + 1}{7}

7 \cdot 7 + 1 = 50

7 \cdot 7 + 1

Multiply the numbers:

7 \cdot 7 = 49

= 49 + 1

Add the numbers:

49 + 1 = 50

= 50

= \frac{50}{7}

= \frac{\frac{50}{7}}{\frac{48}{7}}

Divide fractions:

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

= \frac{50 \cdot 7}{7 \cdot 48}

Cancel the common factor:

7

= \frac{50}{48}

Cancel the common factor:

2

= \frac{25}{24}

= \frac{25}{24}

= \frac{25}{24}

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Frequently Asked Questions (FAQ)

What is the value of coth(ln(7)) ?
The value of coth(ln(7)) is 25/24