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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

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

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

3 ln (4) + 3 ln (3)

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ln (\frac{1}{2} x^{0.5} \cdot y^{0.5})

lo g_{10} (\frac{x ^{5}}{( x + 6 ) ^{3}})

ln (e^{x} cos (x))

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