What is ln(1/2 x^{0.5}*y^{0.5}) ?

The solution to ln(1/2 x^{0.5}*y^{0.5}) is 0.5ln(x)+0.5ln(y)-ln(2)

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simplify ln(1/2 x^{0.5}*y^{0.5})

simplify

ln (\frac{1}{2} x^{0.5} \cdot y^{0.5})

0.5 ln (x) + 0.5 ln (y) - ln (2)

Solution steps

ln (\frac{1}{2} x^{0.5} y^{0.5})

Apply log rule:

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

ln (\frac{1}{2} x^{0.5} y^{0.5}) = ln (\frac{1}{2}) + ln (x^{0.5}) + ln (y^{0.5})

= ln (\frac{1}{2}) + ln (x^{0.5}) + ln (y^{0.5})

Apply log rule:

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

ln (x^{0.5}) = 0.5 ln (x)

= ln (\frac{1}{2}) + 0.5 ln (x) + ln (y^{0.5})

Apply log rule:

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

ln (y^{0.5}) = 0.5 ln (y)

= ln (\frac{1}{2}) + 0.5 ln (x) + 0.5 ln (y)

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)

= ln (1) - ln (2) + 0.5 ln (x) + 0.5 ln (y)

Apply log rule:

lo g_{a} (1) = 0

= 0 - ln (2) + 0.5 ln (x) + 0.5 ln (y)

0 - ln (2) + 0.5 ln (x) + 0.5 ln (y) = - ln (2) + 0.5 ln (x) + 0.5 ln (y)

= 0.5 ln (x) + 0.5 ln (y) - ln (2)

lo g_{10} (\frac{x ^{5}}{( x + 6 ) ^{3}})

ln (e^{x} cos (x))

- 2 ln (x) + ln (y)

3 lo g_{b} (x) - 4 lo g_{b} (y) + \frac{1}{2} lo g_{b} (x + y)

ln (\frac{10}{8})

What is ln(1/2 x^{0.5}*y^{0.5}) ?
The solution to ln(1/2 x^{0.5}*y^{0.5}) is 0.5ln(x)+0.5ln(y)-ln(2)