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

The solution to 4(log_{a}(x)-5log_{a}(y))+3log_{a}(z) is log_{a}((x^4z^3)/(y^{20)})

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simplify 4(log_{a}(x)-5log_{a}(y))+3log_{a}(z)

simplify

4 (lo g_{a} (x) - 5 lo g_{a} (y)) + 3 lo g_{a} (z)

lo g_{a} (\frac{x ^{4} z ^{3}}{y ^{20}})

Solution steps

4 (lo g_{a} (x) - 5 lo g_{a} (y)) + 3 lo g_{a} (z)

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

(\frac{x}{y ^{5}})^{4} z^{3} : \frac{x ^{4} z ^{3}}{y ^{20}}

= lo g_{a} (\frac{x ^{4} z ^{3}}{y ^{20}})

- 3 ln ∣ x + 2 ∣ + x - ln ∣ x - 2 ∣

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

ln (\frac{e ^{X^{2}} cos ( X )}{2 X})

\frac{1}{2} (ln (6) + ln (2))

lo g_{e} (\frac{( 4300 )}{3})

What is 4(log_{a}(x)-5log_{a}(y))+3log_{a}(z) ?
The solution to 4(log_{a}(x)-5log_{a}(y))+3log_{a}(z) is log_{a}((x^4z^3)/(y^{20)})