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

The solution to ln(r)-1/2 ln(r^2+1) is ln((rsqrt(r^2+1))/(r^2+1))

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simplify ln(r)-1/2 ln(r^2+1)

simplify

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

ln (\frac{r r ^{2} + 1}{r ^{2} + 1})

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

\frac{r}{( r ^{2} + 1 ) ^{\frac{1}{2}}} : \frac{r r ^{2} + 1}{r ^{2} + 1}

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

210 ln (2) + 210 ln (3 x - 2 y)

- ln (\frac{2}{2}) \cdot \frac{2}{2}

- [\frac{2}{3} lo g_{2} (\frac{2}{3}) + \frac{1}{3} lo g_{2} (\frac{1}{3})]

2 (lo g_{6} (y) - 2 lo g_{6} (w)) + 2 lo g_{6} (x)

ln (\frac{\frac{10}{3}}{3 \cdot 3.1})

What is ln(r)-1/2 ln(r^2+1) ?
The solution to ln(r)-1/2 ln(r^2+1) is ln((rsqrt(r^2+1))/(r^2+1))