What is ln(r)-1/2 ln(r^2+9) ?

The solution to ln(r)-1/2 ln(r^2+9) is ln((rsqrt(r^2+9))/(r^2+9))

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simplify ln(r)-1/2 ln(r^2+9)

simplify

ln (r) - \frac{1}{2} ln (r^{2} + 9)

ln (\frac{r r ^{2} + 9}{r ^{2} + 9})

Solution steps

ln (r) - \frac{1}{2} ln (r^{2} + 9)

Apply log rule:

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

\frac{1}{2} ln (r^{2} + 9) = ln ((r^{2} + 9)^{\frac{1}{2}})

= ln (r) - ln ((r^{2} + 9)^{\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} + 9)^{\frac{1}{2}}) = ln (\frac{r}{( r ^{2} + 9 ) ^{\frac{1}{2}}})

= ln (\frac{r}{( r ^{2} + 9 ) ^{\frac{1}{2}}})

Simplify

\frac{r}{( r ^{2} + 9 ) ^{\frac{1}{2}}} : \frac{r r ^{2} + 9}{r ^{2} + 9}

= ln (\frac{r r ^{2} + 9}{r ^{2} + 9})

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What is ln(r)-1/2 ln(r^2+9) ?
The solution to ln(r)-1/2 ln(r^2+9) is ln((rsqrt(r^2+9))/(r^2+9))