What is expand ln(\sqrt[5]{5r^4s}) ?

The solution to expand ln(\sqrt[5]{5r^4s}) is 1/5 ln(5s)+4/5 ln(r)

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expand ln(\sqrt[5]{5r^4s})

expand

\frac{1}{5} ln (5 s) + \frac{4}{5} ln (r)

Solution steps

Rewrite as

= ln ((5 r^{4} s)^{\frac{1}{5}})

Apply log rule

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

assuming

x \geq 0

= \frac{1}{5} ln (5 r^{4} s)

Apply log rule:

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

ln (5 r^{4} s) = ln (5 s) + ln (r^{4})

= \frac{1}{5} (ln (r^{4}) + ln (5 s))

Apply log rule

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

assuming

x \geq 0

= \frac{1}{5} (4 ln (r) + ln (5 s))

Apply the distributive law:

a (b + c) = ab + a c

a = \frac{1}{5}, b = ln (5 s), c = 4 ln (r)

= \frac{1}{5} ln (5 s) + \frac{1}{5} \cdot 4 ln (r)

= \frac{1}{5} ln (5 s) + 4 \cdot \frac{1}{5} ln (r)

4 \cdot \frac{1}{5} ln (r) = \frac{4}{5} ln (r)

= \frac{1}{5} ln (5 s) + \frac{4}{5} ln (r)

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What is expand ln(\sqrt[5]{5r^4s}) ?
The solution to expand ln(\sqrt[5]{5r^4s}) is 1/5 ln(5s)+4/5 ln(r)