Happy log day

Algebra Level 3

$\large \ln\left(\dfrac {a+b}3\right) = \dfrac{\ln(a) + \ln(b)}2 \qquad , \qquad \dfrac ab + \dfrac ba = \ ?$

Take $a,b> 0$ .

7 3 1 5

2 solutions

Akhil Bansal
Aug 15, 2015

$\Rightarrow log(\frac{a + b}{3}) = (\frac{log(a) + log(b)}{2})$

$\Rightarrow 2log(\frac{a + b}{3}) = log(ab)$

$\Rightarrow log(\frac{a + b}{3})^2 = (\frac{log(a) + log(b)}{2})$

$\therefore (a +b)^2 = 9ab$
$\Rightarrow a^2 + b^2 + 2ab = 9ab$
$\Rightarrow a^2 + b^2 = 7ab$
$\Rightarrow$ Divide throughout by ab , we get
$\rightarrow \color{#D61F06}{\frac{a}{b} + \frac{b}{a} = 7}$

good one :)

RAJ RAJPUT - 5 years, 10 months ago

Thnx please like and share the problem

Akhil Bansal - 5 years, 10 months ago

Prateek Anand
Aug 16, 2015

Did the same :)

Happy log day

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