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)^{(1)/(4)} \right]\implies ln\left[ x^{(1)/(4)} y^{(1)/(4)} \right] \\\\\\ ln\left( x^{(1)/(4)} \right)+ln\left( y^{(1)/(4)} \right)\implies (1)/(4)ln(x)+(1)/(4)ln(y)`