What is 2ln(x)+ln((x^3)/(y^2))-4ln(sqrt(x+y)) ?

The solution to 2ln(x)+ln((x^3)/(y^2))-4ln(sqrt(x+y)) is ln((x^5)/(y^2(x+y)^2))

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simplify 2ln(x)+ln((x^3)/(y^2))-4ln(sqrt(x+y))

simplify

2 ln (x) + ln (\frac{x ^{3}}{y ^{2}}) - 4 ln (x + y)

ln (\frac{x ^{5}}{y ^{2} ( x + y ) ^{2}})

Solution steps

2 ln (x) + ln (\frac{x ^{3}}{y ^{2}}) - 4 ln (x + y)

Apply log rule:

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

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

= ln (x^{2}) + ln (\frac{x ^{3}}{y ^{2}}) - 4 ln (x + y)

Apply log rule:

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

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

= ln (\frac{x ^{3}}{y ^{2}} x^{2}) - 4 ln (x + y)

Simplify

x^{2} \frac{x ^{3}}{y ^{2}} : \frac{x ^{5}}{y ^{2}}

= ln (\frac{x ^{5}}{y ^{2}}) - 4 ln (x + y)

Apply log rule:

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

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

= ln (\frac{x ^{5}}{y ^{2}}) - ln ((x + y)^{4})

Apply log rule:

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

ln (\frac{x ^{5}}{y ^{2}}) - ln ((x + y)^{4}) = ln (\frac{\frac{x ^{5}}{y ^{2}}}{( x + y ) ^{4}})

= ln (\frac{\frac{x ^{5}}{y ^{2}}}{( x + y ) ^{4}})

Simplify

\frac{\frac{x ^{5}}{y ^{2}}}{( x + y ) ^{4}} : \frac{x ^{5}}{y ^{2} ( x + y ) ^{2}}

= ln (\frac{x ^{5}}{y ^{2} ( x + y ) ^{2}})

ln (x^{3} + 8) - ln (x + 2) - 2 ln (x^{2} - 2 x + 4)

ln (x + 2) - ln (5 x)

lo g_{2} (8 x^{5})

lo g_{b} (x) + lo g_{b} (y) - lo g_{b} (t)

What is 2ln(x)+ln((x^3)/(y^2))-4ln(sqrt(x+y)) ?
The solution to 2ln(x)+ln((x^3)/(y^2))-4ln(sqrt(x+y)) is ln((x^5)/(y^2(x+y)^2))