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

The value of cosh(ln(7)) is 25/7

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cosh(ln(7))

cosh (ln (7))

\frac{25}{7}

Decimal Notation

3.57142 \dots

Solution steps

cosh (ln (7))

Use the Hyperbolic identity:

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

= \frac{e ^{l n (7)} + e ^{- l n (7)}}{2}

\frac{e ^{l n (7)} + e ^{- l n (7)}}{2} = \frac{25}{7}

\frac{e ^{l n (7)} + e ^{- l n (7)}}{2}

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}}{2}

Simplify

\frac{7 + 7 ^{- 1}}{2}

Apply exponent rule:

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

= \frac{7 + \frac{1}{7}}{2}

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}}{2}

Apply the fraction rule:

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

= \frac{50}{7 \cdot 2}

Multiply the numbers:

7 \cdot 2 = 14

= \frac{50}{14}

Cancel the common factor:

2

= \frac{25}{7}

= \frac{25}{7}

= \frac{25}{7}

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