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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

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

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

ln (x^{2} \frac{x - 6}{2 x + 5})

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ln (\frac{11}{41})

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