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

The solution to 1/2 ln(2v^2+1)-ln(v^2+1) is ln(((2v^2+1)^{1/2})/(v^2+1))

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

3 lo g_{b} (q) + 6 lo g_{b} (v)

lo g_{10} (\frac{a b}{c ^{6}})

\frac{1}{2} lo g_{2} (a) + lo g_{2} (b) - 2 lo g_{2} (c)

ln (\frac{0.003637}{0.002967})

lo g_{e} (δ (t))

What is 1/2 ln(2v^2+1)-ln(v^2+1) ?
The solution to 1/2 ln(2v^2+1)-ln(v^2+1) is ln(((2v^2+1)^{1/2})/(v^2+1))