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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

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

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

ln (\frac{729}{625})

lo g_{10} (x^{2} - 5 x + 6) - lo g_{10} (x - 3)

lo g_{6} (a b^{3} c^{- 2})

2 lo g_{2^{x}} (+ 4 lo g_{2^{y}} (- 6 lo g_{2^{z}} (x)))

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