Question

1/2(logx4+logxy)-3logxz

Answer 1

$\bf log_{{ a}}(xy)\implies log_{{ a}}(x)+log_{{ a}}(y) \\ \quad \\ % Logarithm of rationals log_{{ a}}\left( \frac{x}{y}\right)\implies log_{{ a}}(x)-log_{{ a}}(y) \\ \quad \\ % Logarithm of exponentials log_{{ a}}\left( x^{{ b}} \right)\implies {{ b}}\cdot log_{{ a}}(x)\qquad and\qquad a^{\frac{{ n}}{{ m}}} \implies \sqrt[{ m}]{a^{ n}}\\\\ -----------------------------$

$\bf \cfrac{1}{2}\left[ log_x(4)+log_x(y) \right]-3log_x(z) \\\\\\ \cfrac{1}{2}\left[ log_x(4\cdot y) \right]-3log_x(z)\implies log_x\left[ (4y)^{\frac{1}{2}} \right]-log_x(z^3) \\\\\\ log_x\left[ \cfrac{(4y)^{\frac{1}{2}}}{z^3} \right]\implies log_x\left[ \cfrac{\sqrt{4y}}{z^3} \right]$