What is ln(((xy)^2)/(z^3)) ?

The solution to ln(((xy)^2)/(z^3)) is 2ln(x)+2ln(y)-3ln(z)

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simplify ln(((xy)^2)/(z^3))

simplify

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

2 ln (x) + 2 ln (y) - 3 ln (z)

Solution steps

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

Apply log rule:

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

ln (\frac{( x y ) ^{2}}{z ^{3}}) = ln ((x y)^{2}) - ln (z^{3})

= ln ((x y)^{2}) - ln (z^{3})

Apply log rule:

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

ln ((x y)^{2}) = 2 ln (x y)

= 2 ln (x y) - ln (z^{3})

Apply log rule:

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

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

= 2 ln (x y) - 3 ln (z)

Apply log rule:

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

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

= 2 ln (x) + 2 ln (y) - 3 ln (z)

9 ln (t) + ln (y) - ln (y + 1)

lo g_{3} (\frac{c}{9})

lo g_{2} (\frac{1}{3}) + lo g_{4} (9)

lo g_{10} (\frac{2 x}{1 + x ^{2}})

lo g_{b} (\frac{a}{h})

What is ln(((xy)^2)/(z^3)) ?
The solution to ln(((xy)^2)/(z^3)) is 2ln(x)+2ln(y)-3ln(z)