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

The solution to 1/2 ln(1+(y/x)^2)+ln(x) is ln(sqrt(x^2+y^2))

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

simplify

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

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

Solution steps

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

Simplify

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

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

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

ln (θ e^{- n (x^{1} + x^{2} + x^{3})})

ln (\frac{225 - 0.9}{( 2.82108 - 0.9 )})

- \frac{1}{3} ln (\frac{5 + 1}{5 - 2})

ln (\frac{x ^{3}}{y ^{2}})

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