What is the value of 5/2 cosh(ln(5)) ?

The value of 5/2 cosh(ln(5)) is 13/2

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5/2 cosh(ln(5))

\frac{5}{2} cosh (ln (5))

\frac{13}{2}

Decimal Notation

6.5

Solution steps

\frac{5}{2} cosh (ln (5))

Rewrite using trig identities:

cosh (ln (5)) = \frac{13}{5}

cosh (ln (5))

Use the Hyperbolic identity:

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

= \frac{e ^{l n (5)} + e ^{- l n (5)}}{2}

\frac{e ^{l n (5)} + e ^{- l n (5)}}{2} = \frac{13}{5}

\frac{e ^{l n (5)} + e ^{- l n (5)}}{2}

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}}{2}

Simplify

\frac{5 + 5 ^{- 1}}{2}

Apply exponent rule:

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

= \frac{5 + \frac{1}{5}}{2}

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{\frac{26}{5}}{2}

Apply the fraction rule:

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

= \frac{26}{5 \cdot 2}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{26}{10}

Cancel the common factor:

2

= \frac{13}{5}

= \frac{13}{5}

= \frac{13}{5}

= \frac{5}{2} \cdot \frac{13}{5}

Simplify

\frac{5}{2} \cdot \frac{13}{5} : \frac{13}{2}

\frac{5}{2} \cdot \frac{13}{5}

Cross-cancel:

5

= \frac{13}{2}

= \frac{13}{2}

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