What is 2log_{b}(m)-3log_{b}(n)-3log_{b}(p) ?

The solution to 2log_{b}(m)-3log_{b}(n)-3log_{b}(p) is log_{b}((m^2)/(n^3p^3))

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simplify 2log_{b}(m)-3log_{b}(n)-3log_{b}(p)

simplify

2 lo g_{b} (m) - 3 lo g_{b} (n) - 3 lo g_{b} (p)

lo g_{b} (\frac{m ^{2}}{n ^{3} p ^{3}})

Solution steps

2 lo g_{b} (m) - 3 lo g_{b} (n) - 3 lo g_{b} (p)

Apply log rule:

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

2 lo g_{b} (m) = lo g_{b} (m^{2})

= lo g_{b} (m^{2}) - 3 lo g_{b} (n) - 3 lo g_{b} (p)

Apply log rule:

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

3 lo g_{b} (n) = lo g_{b} (n^{3})

= lo g_{b} (m^{2}) - lo g_{b} (n^{3}) - 3 lo g_{b} (p)

Apply log rule:

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

lo g_{b} (m^{2}) - lo g_{b} (n^{3}) = lo g_{b} (\frac{m ^{2}}{n ^{3}})

= lo g_{b} (\frac{m ^{2}}{n ^{3}}) - 3 lo g_{b} (p)

Apply log rule:

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

3 lo g_{b} (p) = lo g_{b} (p^{3})

= lo g_{b} (\frac{m ^{2}}{n ^{3}}) - lo g_{b} (p^{3})

Apply log rule:

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

lo g_{b} (\frac{m ^{2}}{n ^{3}}) - lo g_{b} (p^{3}) = lo g_{b} (\frac{\frac{m ^{2}}{n ^{3}}}{p ^{3}})

= lo g_{b} (\frac{\frac{m ^{2}}{n ^{3}}}{p ^{3}})

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= lo g_{b} (\frac{m ^{2}}{n ^{3} p ^{3}})

- ln ∣ v + 1 ∣ + ln ∣ v - 1 ∣ - \frac{1}{2} ln v^{2} - 1 + C

lo g_{a} (3^{2} 5)

lo g_{3} (H) - 2 lo g_{3} (S) + 5 lo g_{3} (B)

lo g_{4} (\frac{x ^{2} y}{z w ^{3}})

What is 2log_{b}(m)-3log_{b}(n)-3log_{b}(p) ?
The solution to 2log_{b}(m)-3log_{b}(n)-3log_{b}(p) is log_{b}((m^2)/(n^3p^3))