What is ((-1/2 ln(x)-1/4 ln(2+x^2)))/e ?

The solution to ((-1/2 ln(x)-1/4 ln(2+x^2)))/e is -(ln(x^{1/2}(2+x^2)^{1/4}))/e

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simplify ((-1/2 ln(x)-1/4 ln(2+x^2)))/e

simplify

\frac{( - \frac{1}{2} ln ( x ) - \frac{1}{4} ln ( 2 + x ^{2} ) )}{e}

- \frac{ln ( x ^{\frac{1}{2}} ( 2 + x ^{2} ) ^{\frac{1}{4}} )}{e}

Solution steps

\frac{( - \frac{1}{2} ln ( x ) - \frac{1}{4} ln ( 2 + x ^{2} ) )}{e}

Apply log rule:

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

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

= \frac{( - ln ( x ^{\frac{1}{2}} ) - \frac{1}{4} ln ( x ^{2} + 2 ) )}{e}

Apply log rule:

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

\frac{1}{4} ln (x^{2} + 2) = ln ((x^{2} + 2)^{\frac{1}{4}})

= \frac{( - ln ( x ^{\frac{1}{2}} ) - ln ( ( x ^{2} + 2 ) ^{\frac{1}{4}} ) )}{e}

Rewrite

- ln (x^{\frac{1}{2}}) - ln ((x^{2} + 2)^{\frac{1}{4}})

- (ln (x^{\frac{1}{2}}) + ln ((x^{2} + 2)^{\frac{1}{4}}))

= \frac{( - ( ln ( x ^{\frac{1}{2}} ) + ln ( ( x ^{2} + 2 ) ^{\frac{1}{4}} ) ) )}{e}

Apply log rule:

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

ln (x^{\frac{1}{2}}) + ln ((2 + x^{2})^{\frac{1}{4}}) = ln (x^{\frac{1}{2}} (x^{2} + 2)^{\frac{1}{4}})

= \frac{- ln ( x ^{\frac{1}{2}} ( 2 + x ^{2} ) ^{\frac{1}{4}} )}{e}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{ln ( x ^{\frac{1}{2}} ( 2 + x ^{2} ) ^{\frac{1}{4}} )}{e}

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What is ((-1/2 ln(x)-1/4 ln(2+x^2)))/e ?
The solution to ((-1/2 ln(x)-1/4 ln(2+x^2)))/e is -(ln(x^{1/2}(2+x^2)^{1/4}))/e