Question

Given $\log 3=.4771213$, find $\log \left\{\right(2.7{)}^3\times (8.1{)}^{\frac{4}{5}}÷(90{)}^{\frac{5}{4}}\}$.

Answer 1

$\log \{{\left(2.7\right)}^3\times {\left(8.1\right)}^{\frac{4}{5}}÷{\left(90\right)}^{\frac{5}{4}}\}$

$=\log \{{\left(\frac{27}{10}\right)}^3\times {\left(\frac{81}{10}\right)}^{\frac{4}{5}}÷{\left(90\right)}^{\frac{5}{4}}\}$

$=3\log \frac{27}{10}+\frac{4}{5}\log \frac{81}{10}-\frac{5}{4}\log 90$

$=3\left(\log 3^3-1\right)+\frac{4}{5}\left(\log 3^4-1\right)-\frac{5}{4}\left(\log 3^2+1\right)$

$=(9+\frac{16}{5}-\frac{5}{2})\log 3-3-\frac{4}{5}-\frac{5}{4}$

$=\frac{97}{10}\log 3-\frac{101}{20}$

$=\frac{97}{10}\left(0.4771213\right)-\frac{101}{20}$

$=-0.42192339$