Question

Use the laws of logarithms to rewrite the expression ln 4throot(xy) in terms of ln x and ln y. After rewriting ln 4throot(xy) = A ln x + B ln y we find A= and B=

Answer 1

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

$\bf ln\left( \sqrt[4]{xy} \right)\implies ln\left[ (xy)^{\frac{1}{4}} \right]\implies ln\left[ x^{\frac{1}{4}} y^{\frac{1}{4}} \right] \\\\\\ ln\left( x^{\frac{1}{4}} \right)+ln\left( y^{\frac{1}{4}} \right)\implies \frac{1}{4}ln(x)+\frac{1}{4}ln(y)$