What is ln(3)(-1-1/2-1/8+3/4)+ln(2)(-1-1/4-3/4) ?

The solution to ln(3)(-1-1/2-1/8+3/4)+ln(2)(-1-1/4-3/4) is ln((\sqrt[8]{3})/(12))

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simplify ln(3)(-1-1/2-1/8+3/4)+ln(2)(-1-1/4-3/4)

simplify

ln (3) (- 1 - \frac{1}{2} - \frac{1}{8} + \frac{3}{4}) + ln (2) (- 1 - \frac{1}{4} - \frac{3}{4})

Decimal Notation

- 2.34758 \dots

Solution steps

ln (3) (- 1 - \frac{1}{2} - \frac{1}{8} + \frac{3}{4}) + ln (2) (- 1 - \frac{1}{4} - \frac{3}{4})

Apply log rule:

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

ln (3) (\frac{3}{4} - 1 - \frac{1}{2} - \frac{1}{8}) = ln (3^{(\frac{3}{4} - 1 - \frac{1}{2} - \frac{1}{8})})

= ln (3^{(\frac{3}{4} - 1 - \frac{1}{2} - \frac{1}{8})}) + ln (2) (- 1 - \frac{1}{4} - \frac{3}{4})

Apply log rule:

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

ln (2) (- 1 - \frac{1}{4} - \frac{3}{4}) = ln (2^{(- 1 - \frac{1}{4} - \frac{3}{4})})

= ln (3^{(\frac{3}{4} - 1 - \frac{1}{2} - \frac{1}{8})}) + ln (2^{(- 1 - \frac{1}{4} - \frac{3}{4})})

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

ln (3^{(- 1 - \frac{1}{2} - \frac{1}{8} + \frac{3}{4})}) + ln (2^{(- 1 - \frac{1}{4} - \frac{3}{4})}) = ln (3^{\frac{3}{4} - 1 - \frac{1}{2} - \frac{1}{8}} \cdot 2^{- 1 - \frac{1}{4} - \frac{3}{4}})

= ln (3^{\frac{3}{4} - 1 - \frac{1}{2} - \frac{1}{8}} \cdot 2^{- 1 - \frac{1}{4} - \frac{3}{4}})

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What is ln(3)(-1-1/2-1/8+3/4)+ln(2)(-1-1/4-3/4) ?
The solution to ln(3)(-1-1/2-1/8+3/4)+ln(2)(-1-1/4-3/4) is ln((\sqrt[8]{3})/(12))