2
$\begingroup$

I need to compare those two expressions and decide which is bigger.

$2 \sqrt2$ or $\log_2(3)+\log_3(4) $.

So I tried to simplify so the log expression so I know

and so

$$ \log_2(4) \times (\log_4(3) + \log_3(2)) ?? 2 \times \sqrt2$$

and then

$$2 \times \log_2(2)\times(\log_4(3)+\log_3(2)) ?? 2 \sqrt2$$

$$\log_2(2) \times (\log_4(3)+\log_3(2)) ?? \sqrt2 $$

and I know $\ log_2(2) = 1$ so now I need to compare those two expressions:

$$\log_4(3)+\log_3(2) $$against$$ \sqrt2 $$

I'm not really sure what i'm doing wrong here.

$\endgroup$
2
$\begingroup$

$\log_3 4 = \dfrac{\log_2 4}{\log_2 3} = \dfrac{2}{\log_2 3}$

$A := {\log_2 3}+ \log_3 4 = {\log_2 3} + \dfrac{2}{\log_2 3}$

Dividing by$A$ by $\sqrt 2$, observe $ \dfrac{\log_2 3}{\sqrt 2} + \dfrac{\sqrt 2}{\log_2 3} > 2$ by AM-GM inequality (since ${\log_2 3 \ne \sqrt 2}$)

Thus $A>2\sqrt 2$

$\endgroup$
1
$\begingroup$

We have \begin{eqnarray*} ((\ln 3)^2 -2 (\ln 2)^2)^2 \geq 0 \\ ((\ln 3)^4 - 4 ((\ln 3)^2(\ln 2)^2 + 4(\ln 2)^4 \geq 0 \\ ((\ln 3)^4 + 4 ((\ln 3)^2(\ln 2)^2 + 4(\ln 2)^4 \geq 8 ((\ln 3)^2(\ln 2)^2 \\ \end{eqnarray*} Now sqauare root this ... \begin{eqnarray*} ((\ln 3)^2 +2 (\ln 2)^2 \geq \sqrt{8} (\ln 3)(\ln 2) \\ \frac{ \ln 3}{\ln 2}+ \frac{ 2 \ln 2}{ \ln 3} \geq \sqrt{8} \\ \log_2 3 + \log_3 4 \geq 2 \sqrt{2}. \end{eqnarray*}

$\endgroup$
1
$\begingroup$

$$\log_23+\log_34>2\sqrt2$$ because it's $$\log_23+\frac{2}{\log_23}>2\sqrt2,$$ which is AM-GM: $\log_23+\frac{2}{\log_23}>2\sqrt{\log_23\cdot\frac{2}{\log_23}}=2\sqrt2$ or $$\log^2_23-2\sqrt2\log_23+2>0$$ 0r $$\left(\log_23-\sqrt2\right)^2>0,$$ which is obvious.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.