What is 3ln(x^2y)-2ln(xy)+1 ?

The solution to 3ln(x^2y)-2ln(xy)+1 is ln(x^4y)+1

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simplify 3ln(x^2y)-2ln(xy)+1

simplify

3 ln (x^{2} y) - 2 ln (x y) + 1

ln (x^{4} y) + 1

Solution steps

3 ln (x^{2} y) - 2 ln (x y) + 1

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

\frac{( x ^{2} y ) ^{3}}{( x y ) ^{2}} : x^{4} y

= ln (x^{4} y) + 1

ln (x) - 4 ln (x^{2} + 1)

ln (3) - ln (\frac{7}{3})

ln (\frac{0.32}{12})

ln (45) + ln (15)

4 ln (3) + ln (2^{5})