Answer
-.125
Work Step by Step
We are given that $log_{b}2=.43$ and that $ log_{b}3=.68$.
The quotient property of logarithms tells us that $log_{b}\frac{x}{y}=log_{b}x-log_{b}y$ (where x, y, and, b are positive real numbers and $b\ne1$).
Therefore, $log_{b}\sqrt(\frac{2}{3})= log_{b}\frac{\sqrt 2}{\sqrt 3}= log_{b}\sqrt 2-log_{b}\sqrt 3= log_{b}2^{\frac{1}{2}}-log_{b} 3^{\frac{1}{2}}$
The power property of logarithms tells us that $log_{b}x^{r}=r log_{b}x$ (where x and b are positive real numbers, $b\ne1$, and r is a real number).
$log_{b}2^{\frac{1}{2}}-log_{b} 3^{\frac{1}{2}}=\frac{1}{2}log_{b}2-\frac{1}{2}log_{b}3=\frac{1}{2}\times.43-\frac{1}{2}\times.68=.215-.34=-.125$.
