What is the value of sinh(ln(8)) ?

The value of sinh(ln(8)) is 63/16

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sinh(ln(8))

sinh (ln (8))

\frac{63}{16}

Decimal Notation

3.9375

Solution steps

sinh (ln (8))

Simplify:

ln (8) = 3 ln (2)

ln (8)

Rewrite

8

in power-base form:

8 = 2^{3}

= ln (2^{3})

Apply log rule:

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

ln (2^{3}) = 3 ln (2)

= 3 ln (2)

= sinh (3 ln (2))

Rewrite using trig identities:

\frac{63}{16}

sinh (3 ln (2))

Use the Hyperbolic identity:

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

= \frac{e ^{3 l n (2)} - e ^{- 3 l n (2)}}{2}

\frac{e ^{3 l n (2)} - e ^{- 3 l n (2)}}{2} = \frac{63}{16}

\frac{e ^{3 l n (2)} - e ^{- 3 l n (2)}}{2}

e^{3 l n (2)} = 2^{3}

e^{3 l n (2)}

Apply exponent rule:

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

= (e^{l n (2)})^{3}

Apply log rule:

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

e^{l n (2)} = 2

= 2^{3}

e^{- 3 l n (2)} = 2^{- 3}

e^{- 3 l n (2)}

Apply exponent rule:

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

= (e^{l n (2)})^{- 3}

Apply log rule:

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

e^{l n (2)} = 2

= 2^{- 3}

= \frac{2 ^{3} - 2 ^{- 3}}{2}

Simplify

\frac{2 ^{3} - 2 ^{- 3}}{2}

2^{3} = 8

= \frac{8 - 2 ^{- 3}}{2}

2^{- 3} = \frac{1}{8}

2^{- 3}

Apply exponent rule:

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

= \frac{1}{2 ^{3}}

2^{3} = 8

= \frac{1}{8}

= \frac{8 - \frac{1}{8}}{2}

Join

8 - \frac{1}{8} : \frac{63}{8}

8 - \frac{1}{8}

Convert element to fraction:

8 = \frac{8 \cdot 8}{8}

= \frac{8 \cdot 8}{8} - \frac{1}{8}

Since the denominators are equal, combine the fractions:

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

= \frac{8 \cdot 8 - 1}{8}

8 \cdot 8 - 1 = 63

8 \cdot 8 - 1

Multiply the numbers:

8 \cdot 8 = 64

= 64 - 1

Subtract the numbers:

64 - 1 = 63

= 63

= \frac{63}{8}

= \frac{\frac{63}{8}}{2}

Apply the fraction rule:

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

= \frac{63}{8 \cdot 2}

Multiply the numbers:

8 \cdot 2 = 16

= \frac{63}{16}

= \frac{63}{16}

= \frac{63}{16}

= \frac{63}{16}

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