This question has come in DUET math Olympiad 2012
1.If 3\[{3^a}={4^{b}}= 36\]
Then find the value of \[2/a+1/b\]
DUET math olympiad 2012
- Fahim Shahriar
- Posts:138
- Joined:Sun Dec 18, 2011 12:53 pm
Re: DUET math olympiad 2012
\[3^{a}=36\]
\[\Rightarrow log3^{a} = log 36\]
\[\Rightarrow a log3 = log 36\]
\[\Rightarrow a = \frac{log 36}{log 3}\]
Similarly, \
Now \[\frac{2}{a} + \frac{1}{b}\]
\[\frac{2}{\frac{log36}{log3}} + \frac{1}{\frac{log36}{log4}}\]
\[\frac{2log3}{log36} + \frac{log4}{log36}\]
\[\frac{2log3+log4}{log36}\]
\[\frac{log3^{2}+log4}{log36}\]
\[\frac{log(3^{2}*4)}{log36}\]
\[\frac{log36}{log36}\]
\[=1 <====ANSWER\]
\[\Rightarrow log3^{a} = log 36\]
\[\Rightarrow a log3 = log 36\]
\[\Rightarrow a = \frac{log 36}{log 3}\]
Similarly, \
Now \[\frac{2}{a} + \frac{1}{b}\]
\[\frac{2}{\frac{log36}{log3}} + \frac{1}{\frac{log36}{log4}}\]
\[\frac{2log3}{log36} + \frac{log4}{log36}\]
\[\frac{2log3+log4}{log36}\]
\[\frac{log3^{2}+log4}{log36}\]
\[\frac{log(3^{2}*4)}{log36}\]
\[\frac{log36}{log36}\]
\[=1 <====ANSWER\]
Name: Fahim Shahriar Shakkhor
Notre Dame College
Notre Dame College
Re: DUET math olympiad 2012
Essentially the same solution, without using logarithms:
\[\left\{\begin{align*}3^a=36&\Rightarrow 36^{\frac 2a}=3^2\\
4^b=36&\Rightarrow 36^{\frac 1b}=4\end{align*}\right\}\Rightarrow 36^{\frac 2a+\frac 1b}=3^2\times 4=36\Rightarrow \frac 2a+\frac 1b=1.\]
\[\left\{\begin{align*}3^a=36&\Rightarrow 36^{\frac 2a}=3^2\\
4^b=36&\Rightarrow 36^{\frac 1b}=4\end{align*}\right\}\Rightarrow 36^{\frac 2a+\frac 1b}=3^2\times 4=36\Rightarrow \frac 2a+\frac 1b=1.\]
"Everything should be made as simple as possible, but not simpler." - Albert Einstein
- Sazid Akhter Turzo
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Re: DUET math olympiad 2012
@Nayel vaia,
Nice solution .
Turzo
Nice solution .
Turzo
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Re: DUET math olympiad 2012
Mbl theke kisu bujha jai na. Help koren vaia...