What is 5log_{b}(x)+4log_{b}(y)-1/3 log_{b}(z) ?

The solution to 5log_{b}(x)+4log_{b}(y)-1/3 log_{b}(z) is log_{b}((x^5y^4z^{2/3})/z)

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simplify 5log_{b}(x)+4log_{b}(y)-1/3 log_{b}(z)

simplify

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

lo g_{b} (\frac{x ^{5} y ^{4} z ^{\frac{2}{3}}}{z})

Solution steps

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

Apply log rule:

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

5 lo g_{b} (x) = lo g_{b} (x^{5})

= lo g_{b} (x^{5}) + 4 lo g_{b} (y) - \frac{1}{3} lo g_{b} (z)

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^{5}) + lo g_{b} (y^{4}) - \frac{1}{3} lo g_{b} (z)

Apply log rule:

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

lo g_{b} (x^{5}) + lo g_{b} (y^{4}) = lo g_{b} (x^{5} y^{4})

= lo g_{b} (x^{5} y^{4}) - \frac{1}{3} lo g_{b} (z)

Apply log rule:

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

\frac{1}{3} lo g_{b} (z) = lo g_{b} (z^{\frac{1}{3}})

= lo g_{b} (x^{5} y^{4}) - lo g_{b} (z^{\frac{1}{3}})

Apply log rule:

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

lo g_{b} (x^{5} y^{4}) - lo g_{b} (z^{\frac{1}{3}}) = lo g_{b} (\frac{x ^{5} y ^{4}}{z ^{\frac{1}{3}}})

= lo g_{b} (\frac{x ^{5} y ^{4}}{z ^{\frac{1}{3}}})

Simplify

\frac{x ^{5} y ^{4}}{z ^{\frac{1}{3}}} : \frac{x ^{5} y ^{4} z ^{\frac{2}{3}}}{z}

= lo g_{b} (\frac{x ^{5} y ^{4} z ^{\frac{2}{3}}}{z})

lo g_{3} (x - 1) + lo g_{3} (2 x + 3)

\frac{ln ( \frac{1024}{5} )}{2 ln ( 2 )}

ln (e^{2 x} (1 + x^{3}))

lo g_{10} (\frac{1}{2 π x ^{2}})

ln (2 c d^{3})

What is 5log_{b}(x)+4log_{b}(y)-1/3 log_{b}(z) ?
The solution to 5log_{b}(x)+4log_{b}(y)-1/3 log_{b}(z) is log_{b}((x^5y^4z^{2/3})/z)