Question

Question #12 Please help

Answer 1

`\bf \textit{logarithm of factors}\\\\ log_a(xy)\implies log_a(x)+log_a(y) \\\\\\ \textit{Logarithm of rationals}\\\\ log_a\left( (x)/(y)\right)\implies log_a(x)-log_a(y) \\\\\\ \textit{Logarithm of exponentials}\\\\ log_a\left( x^ b \right)\implies b\cdot log_a(x)\\\\ -------------------------------\\\\`


`\bf a)\\\\ log_9\left( \sqrt[3]{y^2+y} \right)\implies log_9\left[ (y^2+y)^{(1)/(3)} \right]\implies \cfrac{1}{3}log_9(y^2+y) \\\\\\ \cfrac{1}{3}log_9[y(y+1)]\implies \cfrac{1}{3}\left[~log_9(y)~+~log_9(y+1) ~ \right] \\\\\\ \cfrac{1}{3}log_9(y)~~+~~\cfrac{1}{3}log_9(y+1)`



`\bf b)\\\\ log_4\left( (a^2b)/(√(c)) \right)\implies log_4(a^2b)~-~log_4(√(c))\implies log_4(a^2b)~-~log_4(c^{(1)/(2)}) \\\\\\\ [~log_4(a^2)~+log_4(b)~]~~-~~\cfrac{1}{2}log_4(c)\\\\\\ log_4(a^2)+log_4(b)-\cfrac{1}{2}log_4(c)`