What is 3log_{b}(x)-4log_{b}(y)+1/2 log_{b}(x+y) ?

The solution to 3log_{b}(x)-4log_{b}(y)+1/2 log_{b}(x+y) is log_{b}((x^3(x+y)^{1/2})/(y^4))

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simplify 3log_{b}(x)-4log_{b}(y)+1/2 log_{b}(x+y)

simplify

3 lo g_{b} (x) - 4 lo g_{b} (y) + \frac{1}{2} lo g_{b} (x + y)

lo g_{b} (\frac{x ^{3} ( x + y ) ^{\frac{1}{2}}}{y ^{4}})

Solution steps

3 lo g_{b} (x) - 4 lo g_{b} (y) + \frac{1}{2} lo g_{b} (x + y)

Apply log rule:

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

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

= lo g_{b} (x^{3}) - 4 lo g_{b} (y) + \frac{1}{2} lo g_{b} (x + y)

Apply log rule:

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

4 lo g_{b} (y) = lo g_{b} (y^{4})

= lo g_{b} (x^{3}) - lo g_{b} (y^{4}) + \frac{1}{2} lo g_{b} (x + y)

Apply log rule:

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

lo g_{b} (x^{3}) - lo g_{b} (y^{4}) = lo g_{b} (\frac{x ^{3}}{y ^{4}})

= lo g_{b} (\frac{x ^{3}}{y ^{4}}) + \frac{1}{2} lo g_{b} (x + y)

Apply log rule:

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

\frac{1}{2} lo g_{b} (x + y) = lo g_{b} ((x + y)^{\frac{1}{2}})

= lo g_{b} (\frac{x ^{3}}{y ^{4}}) + lo g_{b} ((x + y)^{\frac{1}{2}})

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

lo g_{b} (\frac{x ^{3}}{y ^{4}}) + lo g_{b} ((x + y)^{\frac{1}{2}}) = lo g_{b} (\frac{x ^{3}}{y ^{4}} (x + y)^{\frac{1}{2}})

= lo g_{b} (\frac{x ^{3}}{y ^{4}} (x + y)^{\frac{1}{2}})

Multiply fractions:

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

= lo g_{b} (\frac{x ^{3} ( x + y ) ^{\frac{1}{2}}}{y ^{4}})

ln (\frac{10}{8})

ln (\frac{9}{13})

4 lo g_{6} (x) - \frac{1}{3} lo g_{6} (y)

lo g_{7} ((x y z^{2})^{3})

5 ln ∣ x + 3 ∣ - \frac{8}{x + 3} + 5 ln ∣ x + 7 ∣

What is 3log_{b}(x)-4log_{b}(y)+1/2 log_{b}(x+y) ?
The solution to 3log_{b}(x)-4log_{b}(y)+1/2 log_{b}(x+y) is log_{b}((x^3(x+y)^{1/2})/(y^4))