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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

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

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

ln (\frac{e ^{y}}{2})

ln (\frac{1}{a + b \cdot \frac{x}{d} + c \cdot ( \frac{x}{d} ) ^{g}})

ln (a + b) - ln (a^{2} - b^{2}) + 2 ln (c)

3 lo g_{10} (2) + 4 lo g_{10} (x)

5 lo g_{a} (x) - 7 lo g_{a} (y^{6})

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