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

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

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

simplify

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

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

Solution steps

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

Rewrite

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

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

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

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

Simplify

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

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

ln ∣ 5 ∣ - 2 ln ∣ 2 ∣

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ln (2 e^{\frac{- π}{4} i})

3 ln (2 e^{1})

ln (∣ x - 1 ∣^{2}) - ln ∣ x - 1 ∣

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