What is the value of cosh(ln(6)) ?

The value of cosh(ln(6)) is 37/12

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cosh(ln(6))

cosh (ln (6))

\frac{37}{12}

Decimal Notation

3.08333 \dots

Solution steps

cosh (ln (6))

Use the Hyperbolic identity:

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

= \frac{e ^{l n (6)} + e ^{- l n (6)}}{2}

\frac{e ^{l n (6)} + e ^{- l n (6)}}{2} = \frac{37}{12}

\frac{e ^{l n (6)} + e ^{- l n (6)}}{2}

e^{l n (6)} = 6

e^{l n (6)}

Apply log rule:

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

= 6

e^{- l n (6)} = 6^{- 1}

e^{- l n (6)}

Apply exponent rule:

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

= (e^{l n (6)})^{- 1}

Apply log rule:

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

e^{l n (6)} = 6

= 6^{- 1}

= \frac{6 + 6 ^{- 1}}{2}

Simplify

\frac{6 + 6 ^{- 1}}{2}

Apply exponent rule:

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

= \frac{6 + \frac{1}{6}}{2}

Join

6 + \frac{1}{6} : \frac{37}{6}

6 + \frac{1}{6}

Convert element to fraction:

6 = \frac{6 \cdot 6}{6}

= \frac{6 \cdot 6}{6} + \frac{1}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{6 \cdot 6 + 1}{6}

6 \cdot 6 + 1 = 37

6 \cdot 6 + 1

Multiply the numbers:

6 \cdot 6 = 36

= 36 + 1

Add the numbers:

36 + 1 = 37

= 37

= \frac{37}{6}

= \frac{\frac{37}{6}}{2}

Apply the fraction rule:

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

= \frac{37}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{37}{12}

= \frac{37}{12}

= \frac{37}{12}

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