What is 2ln(x^3*y 1/2) ?

The solution to 2ln(x^3*y 1/2) is 6ln(x)+2ln(y)-2ln(2)

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simplify 2ln(x^3*y 1/2)

simplify

2 ln (x^{3} \cdot y \frac{1}{2})

6 ln (x) + 2 ln (y) - 2 ln (2)

Solution steps

2 ln (x^{3} y \frac{1}{2})

Apply log rule:

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

2 ln (x^{3} y \frac{1}{2}) = 2 ln (x^{3}) + 2 ln (y) + 2 ln (\frac{1}{2})

= 2 ln (x^{3}) + 2 ln (y) + 2 ln (\frac{1}{2})

Apply log rule:

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

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

= 2 \cdot 3 ln (x) + 2 ln (y) + 2 ln (\frac{1}{2})

Apply log rule:

lo g_{c} (\frac{a}{b}) = lo g_{c} (a) - lo g_{c} (b)

ln (\frac{1}{2}) = ln (1) - ln (2)

= 2 \cdot 3 ln (x) + 2 ln (y) + 2 (ln (1) - ln (2))

2 \cdot 3 ln (x) = 6 ln (x)

2 (ln (1) - ln (2)) = - 2 ln (2)

= 6 ln (x) + 2 ln (y) - 2 ln (2)

ln (\frac{0.19556}{0.1563})

50 \cdot (ln (900) - ln (50))

\frac{ln ( 7.1 ) - ln ( 7 )}{7.1 - 7}

ln (- 3 e^{- 3 x})

- lo g_{10} (2.01 \cdot 1 0^{- 6})