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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

\frac{1}{4} lo g_{b} (x) = lo g_{b} (x^{\frac{1}{4}})

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

Apply log rule:

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

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

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

Simplify

\frac{z ^{2}}{x ^{\frac{1}{4}}} : \frac{z ^{2} x ^{\frac{3}{4}}}{x}

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

\frac{z ^{2} x ^{\frac{3}{4}}}{x} y^{3} : \frac{z ^{2} y ^{3} x ^{\frac{3}{4}}}{x}

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

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

x - 1 + ln (\frac{x - 2}{x + 2})

lo g_{10} (\frac{2 x}{y ^{3}})

ln (\frac{1}{x 2 π})

4 lo g_{2} (y - 3) - 6 lo g_{2} (y + 9)

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