What is ln((y^3*x^2*y^2)/(x^{(1/3))}) ?

The solution to ln((y^3*x^2*y^2)/(x^{(1/3))}) is 5ln(y)+5/3 ln(x)

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simplify ln((y^3x^2y^2)/(x^{(1/3))})

simplify

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

5 ln (y) + \frac{5}{3} ln (x)

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Group like terms

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

Add similar elements:

3 ln (y) + 2 ln (y) = 5 ln (y)

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

Add similar elements:

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

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

\frac{1}{3} ln ∣ x - 3 ∣ + \frac{1}{3} ln x^{2} + 9

ln (2) ln (\frac{2 y}{0.5})

ln (2 e^{x + 1})

ln (\frac{3}{ln ( 4 )})

ln (\frac{5}{1000000})

What is ln((y^3*x^2*y^2)/(x^{(1/3))}) ?
The solution to ln((y^3*x^2*y^2)/(x^{(1/3))}) is 5ln(y)+5/3 ln(x)