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

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

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

simplify

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

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

Solution steps

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

Rewrite

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

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

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

Apply log rule:

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

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

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

Simplify

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

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

- lo g_{10} (0.01 \cdot 1 0^{- 9})

lo g_{2} (57.6) - lo g_{2} (1.8)

- lo g_{10} (3.97 X 1 0^{- 11})

(ln (a + x) - ln (a)) (ln (a + x) + ln (a))

lo g_{e} (\frac{n !}{( n - x ) ! x !} (Y)^{x} (1 - Y)^{n - x})

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