What is (ln(x+yi)-ln(x-yi))/(2i) ?

The solution to (ln(x+yi)-ln(x-yi))/(2i) is -(iln((x^2-y^2+2ixy)/(x^2+y^2)))/2

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

simplify (ln(x+yi)-ln(x-yi))/(2i)

simplify

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

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

Solution steps

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

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (x + i y) - ln (x - i y) = ln (\frac{x + i y}{x - i y})

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

Simplify

\frac{x + y i}{x - y i} : \frac{x ^{2} - y ^{2}}{x ^{2} + y ^{2}} + i \frac{2 x y}{x ^{2} + y ^{2}}

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

Multiply

i \frac{2 x y}{x ^{2} + y ^{2}} : \frac{2 i x y}{x ^{2} + y ^{2}}

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

Combine the fractions

\frac{x ^{2} - y ^{2}}{x ^{2} + y ^{2}} + \frac{2 i x y}{x ^{2} + y ^{2}} : \frac{x ^{2} - y ^{2} + 2 i x y}{x ^{2} + y ^{2}}

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

Rationalize

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

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

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

ln (\frac{2.26}{12})

ln^{2} (\frac{11}{10})

5 ln (\frac{x}{6})

lo g_{10} (0.1) + lo g_{10} (0.05)

e^{5} ln (5 x)

What is (ln(x+yi)-ln(x-yi))/(2i) ?
The solution to (ln(x+yi)-ln(x-yi))/(2i) is -(iln((x^2-y^2+2ixy)/(x^2+y^2)))/2