1. ## Proof of Logarithms

Hi,
I was looking at a math book when I stumbled across a problem:
Prove the following:
$log x < x$ given $x > 0$
I am a little puzzled. How would I prove it? Using proof by counterexample?

Thanks!

2. You can prove this using derivatives...

3. Okay... Is it possible without derivatives?

4. Why would you want a different method?

First note that $\displaystyle \ln{x} < 0$ for all $\displaystyle x \in (0,1)$, while $\displaystyle x > 0$ for all $\displaystyle x > 0$. So $\displaystyle \ln{x} < x$ for all $\displaystyle x \in (0, 1)$.

Then note that $\displaystyle \frac{d}{dx}(\ln{x}) = \frac{1}{x}$ and $\displaystyle \frac{d}{dx}(x) = 1$.

For $\displaystyle x \geq 1$, we have $\displaystyle \frac{1}{x} < 1$

So that means $\displaystyle \ln{x}$ grows slower than $\displaystyle x$ does, so can never catch up.

Therefore $\displaystyle \ln{x} < x$ for all $\displaystyle x > 0$.

5. Okay, that makes sense... So we can use growth rates of functions to determine which is the largest in the end.

6. Use a contradiction: suppose not, that means $\ln x\ge x,$ that would imply $x\ge e^x,$ but this is obviously false since $x>0.$