Question

Ln(r^2s^10 4√r^2s^10) is equal to Alnr+Blns Where A equals and B equals

Answer 1

`\textit{logarithm of factors} \\\\ \log_a(xy)\implies \log_a(x)+\log_a(y) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \ln\left( r^2s^(10)\sqrt[4]{r^2s^(10)} \right)\qquad \qquad \stackrel{\textit{let's for a second make}}{z = r^2s^(10)} \\\\\\ \ln\left(z\cdot \sqrt[4]{z} \right)\implies \ln\left(z\cdot z^{(1)/(4)} \right)\implies \ln\left(z^{1+(1)/(4)} \right)\implies \ln\left(z^{(5)/(4)} \right)`

`\stackrel{\textit{and substituting back}}{\ln\left( \left[ r^2s^(10) \right]^{(5)/(4)} \right)}\implies \ln\left( r^{2\cdot (5)/(4)} ~~ s^{10\cdot (5)/(4)} \right)\implies \ln\left( r^{(5)/(2)}~~s^{(25)/(2)} \right) \\\\\\ \ln\left( r^{(5)/(2)} \right)~~ + ~~\ln\left( s^{(25)/(2)} \right)\implies \stackrel{A}{\cfrac{5}{2}}\ln(r)~~ + ~~\stackrel{B}{\cfrac{25}{2}}\ln(s)`