What is 1/2 ln^2(x)+2ln(x) ?

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

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simplify 1/2 ln^2(x)+2ln(x)

simplify

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

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

Solution steps

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

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

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

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

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

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

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

Apply log rule:

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

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

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

Simplify

x^{\frac{1}{2}} x^{2} : x^{\frac{5}{2}}

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

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