Alg9.4.8

Express $\frac{3}{2}\log(x)-\frac{1}{3}\log(y)+\frac{2}{3}\log(z)-\frac{1}{5}log(a)\,$ as a single logarithm.

Rearranging the given logarithms as per the logarithmic formula, $m\log(a)=\log(a^m)\,$

we get

$\log(x^{\frac{3}{2}})-\log(y^{\frac{1}{3}})+\log(z^{\frac{2}{3}})-\log(a^{\frac{1}{5}})\,$

Further rearranging the terms,it looks like

$\log(x^{\frac{3}{2}})+\log(z^{\frac{2}{3}})-(\log(y^{\frac{1}{3}})+\log(a^{\frac{1}{5}})\,$

Now as per the formula $\log(m)+\log(n)=log(mn)\,$ with the same base

we get

$\log(x^{\frac{3}{2}}z^{\frac{2}{3}})-\log(y^{\frac{1}{3}}a^{\frac{1}{5}})\,$

Again applying the formula $\log(m)-\log(n)=log(\frac{m}{n})\,$ with the same base

we get $\log(\frac{x^{\frac{3}{2}}z^{\frac{2}{3}}}{y^{\frac{1}{3}}a^{\frac{1}{5}}})\,$

Expressing in still the simpler form

$\log(\frac{\sqrt{x^3}\sqrt[3]{z^2}}{\sqrt[3]{y}\sqrt[5]{a}})\,$