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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

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

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

2 ln (+ x^{3} ln (x))

ln (3 + 2 y) - ln (1 + 3 y)

2 ln (2) - 2 ln (3)

\frac{3 ln ( 2 )}{ln ( \frac{3}{2} )}

2 lo g_{10} (0.000001) + 2 lo g_{10} (0.001)

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