Comparing two numbers in symbolic form

Question

How can I compare two numbers in symbolic form? For example, how can I prove, that $ln(2)\times\sqrt5<ln(3)\times\sqrt2$?

Answer 1

Use continued fractions to get a good upper bound on $\sqrt{5}$ and a lower bound on $\sqrt{2}$. \begin{eqnarray*} \sqrt{5}= 2 +\frac{1}{4+\frac{1}{\ddots}} \\ \sqrt{2}= 1 +\frac{1}{2+\frac{1}{2+\frac{1}{\ddots}}} \end{eqnarray*} So $\sqrt{5}<\frac{161}{72}$ and $\frac{41}{29} < \sqrt{2}$.Now Wolfram alpha says $2^{161 \times 29}=3.23\cdots \times 10^{1405}$ and $3^{41 \times 72}=2.90\cdots \times 10^{1408}$ Then \begin{eqnarray*} 2^\sqrt{5}< 2 ^\frac{161}{72} < 3^{\frac{41}{29}}<3^\sqrt{2} \end{eqnarray*} Now take natural logarithms.