What is expand ln(sqrt((x^3y)/(z^5))) ?

The solution to expand ln(sqrt((x^3y)/(z^5))) is 1/2 ln(y)+3/2 ln(x)-5/2 ln(z)

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expand ln(sqrt((x^3y)/(z^5)))

expand

ln (\frac{x ^{3} y}{z ^{5}})

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

Solution steps

ln (\frac{x ^{3} y}{z ^{5}})

Rewrite as

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

Apply log rule

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

assuming

x \geq 0

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

Simplify

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

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

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

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

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

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

Simplify

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

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

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

(x^{2} - 3) \cdot e^{x}

(\frac{x}{3} + \frac{y}{2} = 7) 3

\frac{6}{( 5 x - 4 ) ^{2}}

\frac{1}{x ^{2} ( x + k )}

What is expand ln(sqrt((x^3y)/(z^5))) ?
The solution to expand ln(sqrt((x^3y)/(z^5))) is 1/2 ln(y)+3/2 ln(x)-5/2 ln(z)