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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

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

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

- lo g_{10} (4.61 \cdot 1 0^{- 7})

\frac{2 ln ( 3 )}{ln ( 7 )} - 10

lo g_{10} (3 x^{3}) \cdot ln (10)

ln (\frac{11 π}{2})

ln (x y^{7})

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