$\displaystyle \ln{\sqrt{\frac{8}{27}}}$ | Given |
$\displaystyle = \ln{\left\frac{8}{27}\right} ^\frac{1}{2}$ | Equivalent fractional exponent |
$\displaystyle =\frac{\ln{\frac{8}{27}}}{2}$ | Log property: $\displaystyle \log_b{a^n} = n\log_b{a}$ |
$\displaystyle =\frac{\ln{8} - \ln{27}}{2}$ | Log property: $\displaystyle \log_b{\frac{a}{n}} = \log_b{a} - \log_b{n}$ |
$\displaystyle =\frac{\ln{2^3} - \ln{3^3}}{2}$ | $\displaystyle 8 = 2^3$ and $\displaystyle 27 = 3^3$ |
$\displaystyle =\frac{3\ln{2} - 3\ln{3}}{2}$ | Log property: $\displaystyle \log_b{a^n} = n\log_b{a}$ |
$\displaystyle =\frac{3(0.6931) - 3(1.0986)}{2}$ | Substitute $\displaystyle \ln 2$ and $\displaystyle \ln 3$ |
$\displaystyle \approx -0.60825$ | Simplify with calculator |