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Popular >

expand ln(sqrt(x^5))

Solution

expand

ln (x^{5})

Solution

\frac{5}{2} ln (x)

Solution steps

ln (x^{5})

x^{5} = x^{2} x

= ln (x^{2} x)

Apply log rule:

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

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

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

Apply log rule:

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

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

= ln (x x^{2})

Apply log rule:

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

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

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

Simplify

ln (x) : \frac{1}{2} ln (x)

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

Apply log rule

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

assuming

x \geq 0

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

Factor out common term

ln (x)

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

2 + \frac{1}{2} = \frac{5}{2}

= \frac{5}{2} ln (x)

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Frequently Asked Questions (FAQ)

What is expand ln(sqrt(x^5)) ?
The solution to expand ln(sqrt(x^5)) is 5/2 ln(x)