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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

\frac{1}{5} lo g_{a} (z) = lo g_{a} (z^{\frac{1}{5}})

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

Apply log rule:

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

lo g_{a} (x^{5}) - lo g_{a} (z^{\frac{1}{5}}) = lo g_{a} (\frac{x ^{5}}{z ^{\frac{1}{5}}})

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

Simplify

\frac{x ^{5}}{z ^{\frac{1}{5}}} : \frac{x ^{5} z ^{\frac{4}{5}}}{z}

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

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

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

lo g_{e} (\frac{x ^{2} + 4}{x ^{2} - 4})

ln (3 e^{x})

ln (\frac{3}{81})

3.15 + lo g_{10} (\frac{1.875}{0.625})

\frac{1}{2} (ln (5) - ln (4))

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