What is expand log_{5}(\sqrt[3]{x^2y}) ?

The solution to expand log_{5}(\sqrt[3]{x^2y}) is 1/3 log_{5}(y)+2/3 log_{5}(x)

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expand log_{5}(\sqrt[3]{x^2y})

expand

\frac{1}{3} lo g_{5} (y) + \frac{2}{3} lo g_{5} (x)

Solution steps

Rewrite as

= lo g_{5} ((x^{2} y)^{\frac{1}{3}})

Apply log rule

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

assuming

x \geq 0

= \frac{1}{3} lo g_{5} (x^{2} y)

Simplify

lo g_{5} (x^{2} y) : 2 lo g_{5} (x) + lo g_{5} (y)

= \frac{1}{3} (2 lo g_{5} (x) + lo g_{5} (y))

Apply the distributive law:

a (b + c) = ab + a c

a = \frac{1}{3}, b = lo g_{5} (y), c = 2 lo g_{5} (x)

= \frac{1}{3} lo g_{5} (y) + \frac{1}{3} \cdot 2 lo g_{5} (x)

= \frac{1}{3} lo g_{5} (y) + 2 \cdot \frac{1}{3} lo g_{5} (x)

2 \cdot \frac{1}{3} lo g_{5} (x) = \frac{2}{3} lo g_{5} (x)

= \frac{1}{3} lo g_{5} (y) + \frac{2}{3} lo g_{5} (x)

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(\frac{1}{x ^{3}} - 2 x)^{16}

What is expand log_{5}(\sqrt[3]{x^2y}) ?
The solution to expand log_{5}(\sqrt[3]{x^2y}) is 1/3 log_{5}(y)+2/3 log_{5}(x)