What is 7log_{b}(y)-1/3 log_{b}(x)+5log_{b}(z) ?

The solution to 7log_{b}(y)-1/3 log_{b}(x)+5log_{b}(z) is log_{b}((y^7z^5x^{2/3})/x)

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simplify 7log_{b}(y)-1/3 log_{b}(x)+5log_{b}(z)

simplify

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

lo g_{b} (\frac{y ^{7} z ^{5} x ^{\frac{2}{3}}}{x})

Solution steps

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

Apply log rule:

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

7 lo g_{b} (y) = lo g_{b} (y^{7})

= lo g_{b} (y^{7}) - \frac{1}{3} lo g_{b} (x) + 5 lo g_{b} (z)

Apply log rule:

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

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

= lo g_{b} (y^{7}) - lo g_{b} (x^{\frac{1}{3}}) + 5 lo g_{b} (z)

Apply log rule:

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

lo g_{b} (y^{7}) - lo g_{b} (x^{\frac{1}{3}}) = lo g_{b} (\frac{y ^{7}}{x ^{\frac{1}{3}}})

= lo g_{b} (\frac{y ^{7}}{x ^{\frac{1}{3}}}) + 5 lo g_{b} (z)

Simplify

\frac{y ^{7}}{x ^{\frac{1}{3}}} : \frac{y ^{7} x ^{\frac{2}{3}}}{x}

= lo g_{b} (\frac{y ^{7} x ^{\frac{2}{3}}}{x}) + 5 lo g_{b} (z)

Apply log rule:

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

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

= lo g_{b} (\frac{y ^{7} x ^{\frac{2}{3}}}{x}) + lo g_{b} (z^{5})

Apply log rule:

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

lo g_{b} (\frac{y ^{7} x ^{\frac{2}{3}}}{x}) + lo g_{b} (z^{5}) = lo g_{b} (\frac{y ^{7} x ^{\frac{2}{3}}}{x} z^{5})

= lo g_{b} (\frac{y ^{7} x ^{\frac{2}{3}}}{x} z^{5})

Simplify

\frac{y ^{7} x ^{\frac{2}{3}}}{x} z^{5} : \frac{y ^{7} z ^{5} x ^{\frac{2}{3}}}{x}

= lo g_{b} (\frac{y ^{7} z ^{5} x ^{\frac{2}{3}}}{x})

- lo g_{10} (9 \times 1 0^{- 5})

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

\frac{1}{5} ln (3)

lo g_{\frac{a}{2}} (\frac{a ^{2}}{4})

lo g_{5} (7) + lo g_{5} (2)

What is 7log_{b}(y)-1/3 log_{b}(x)+5log_{b}(z) ?
The solution to 7log_{b}(y)-1/3 log_{b}(x)+5log_{b}(z) is log_{b}((y^7z^5x^{2/3})/x)