Compare an expression with zero

Question

I need to compare $1-\frac{2}{3}\cdot3^{-\frac{2}{3}}\cdot \log_e9$ and $0$ without any computer

Answer 1

With the use of $\;9<e^3$ we get $$\frac{2}{3}\cdot3^{-\frac{2}{3}}\cdot \ln 9<\frac{2}{3}\cdot3^{-\frac{2}{3}}\cdot \ln e^3=\frac{2}{9^{1\over 3}}<\frac{2}{8^{1\over 3}}=1,$$ thus $$1-\frac{2}{3}\cdot3^{-\frac{2}{3}}\cdot \ln 9>0.$$

Answer 2

It's easy to take conservative estimates $3^{-\frac{2}{3}}<\frac{1}{2}$, $\log_e9<\frac{5}{2}$ and conclude that the expression is positive.