Problem: Let be positive reals. Prove that
Solution: How do you get rid of exponents? Well, take the log of course! Logging both sides and using log rules (multiplication to addition and bringing the power down), the inequality becomes
We will prove that
If we add those together, we get the inequality above. But we note that and are similarly sorted, we can apply the Rearrangement Inequality (see here), which tells us that
exactly what we needed. Thus the inequality is proven. QED.
Comment: Classic way of proving inequalities with exponents; many times, after you take the log, you can apply Jensen’s (log is concave) to get some interesting results as well.
Practice Problem: Prove AM-GM. For positive reals ,
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