Question

Expand the log as a sun or difference

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 log_a\left( \cfrac{9^{(1)/(2)}}{x^43^{(1)/(3)}} \right)\implies log_a\left( 9^{(1)/(2)} \right)-log_a\left( x^43^{(1)/(3)} \right) \\\\\\ log_a\left( 9^{(1)/(2)} \right)-\left[ log_a\left( x^4 \right)+log_a\left( 3^{(1)/(3)} \right) \right] \\\\\\ log_a\left( 9^{(1)/(2)} \right)-log_a\left( x^4 \right)-log_a\left( 3^{(1)/(3)} \right)\\\\\\ \cfrac{1}{2}log_a(9)-4log_a(x)-\cfrac{1}{3}log_a(3)`

or more expanded so as


`\bf \cfrac{1}{2}log_a(3^2)-4log_a(x)-\cfrac{1}{3}log_a(3) \\\\\\ \cfrac{1}{2}\cdot 2log_a(3)-4log_a(x)-\cfrac{1}{3}log_a(3) \\\\\\ log_a(3)-4log_a(x)-\cfrac{1}{3}log_a(3)`

Answer 2

27. (1/2*log(9) -(4*log(x) +1/3*log(3))/log(a)
= ((1/2)*2*log(3) -4*log(x) -1/3*log(3))/log(a)
= ((2/3)*log(3) -4*log(x))/log(a)
=
`(2)/(3)\log_a{(3)}-4\log_a{(x)}`