What is 2ln(3x^5y)+1/2 ln(xy^3)-ln(z) ?

The solution to 2ln(3x^5y)+1/2 ln(xy^3)-ln(z) is ln((9x^{10}y^3sqrt(x)sqrt(y))/z)

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simplify 2ln(3x^5y)+1/2 ln(xy^3)-ln(z)

simplify

2 ln (3 x^{5} y) + \frac{1}{2} ln (x y^{3}) - ln (z)

ln (\frac{9 x ^{10} y ^{3} x y}{z})

Solution steps

2 ln (3 x^{5} y) + \frac{1}{2} ln (x y^{3}) - ln (z)

Apply log rule:

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

2 ln (3 x^{5} y) = ln ((3 x^{5} y)^{2})

= ln ((3 x^{5} y)^{2}) + \frac{1}{2} ln (x y^{3}) - ln (z)

Apply log rule:

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

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

= ln ((3 x^{5} y)^{2}) + ln ((x y^{3})^{\frac{1}{2}}) - ln (z)

Apply log rule:

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

ln ((3 x^{5} y)^{2}) + ln ((x y^{3})^{\frac{1}{2}}) = ln ((3 x^{5} y)^{2} (x y^{3})^{\frac{1}{2}})

= ln ((3 x^{5} y)^{2} (x y^{3})^{\frac{1}{2}}) - ln (z)

Simplify

(3 x^{5} y)^{2} (x y^{3})^{\frac{1}{2}} : 9 x^{10} y^{3} x y

= ln (9 x^{10} y^{3} x y) - ln (z)

Apply log rule:

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

ln (9 x^{10} y^{3} x y) - ln (z) = ln (\frac{9 x ^{10} y ^{3} x y}{z})

= ln (\frac{9 x ^{10} y ^{3} x y}{z})

4.74 + lo g_{10} (\frac{2.1245}{3.5255})

ln (\frac{y}{a})

ln (\frac{100000 - 20}{20 ( 50000 )})

lo g_{2} (39 x y^{2}) - lo g_{3} (27 x y)

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

What is 2ln(3x^5y)+1/2 ln(xy^3)-ln(z) ?
The solution to 2ln(3x^5y)+1/2 ln(xy^3)-ln(z) is ln((9x^{10}y^3sqrt(x)sqrt(y))/z)