S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
It is currently Thu, 24 Jul 2014 09:25:22 UTC

All times are UTC [ DST ]




Post new topic Reply to topic  [ 4 posts ] 
Author Message
 Post subject: log inequality
PostPosted: Thu, 17 May 2012 04:32:53 UTC 
Offline
Member

Joined: Sun, 8 Apr 2012 09:36:59 UTC
Posts: 29
How can I prove \displaystyle \frac{1}{3}<\log_{20}(3)<\frac{1}{2}


Top
 Profile  
 
 Post subject: Re: log inequality
PostPosted: Thu, 17 May 2012 05:26:25 UTC 
Offline
Member of the 'S.O.S. Math' Hall of Fame

Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 8106
Location: Lexington, MA
Hello, jacks!

I did it by working backwards . . .


Quote:
How can I prove: .\frac{1}{3}\:<\:\log_{20}(3)\:<\:\frac{1}{2}

We know that: .400 \:<\:729 \:<\:8000 \quad\Rightarrow\quad 20^2 \:<\:3^6 \:<\:20^3


Take logs, base 20: .\log_{20}(20^2) \:<\:\log_{20}(3^6) \:<\:\log_{20}(20^3)

. . . . . . . . . . . . . .2\log_{20}(20) \:<\:6\log_{20}(3) \:<\:3\log_{20}(20)

. . . . . . . . . . . . . . . . . . . . . 2 \:<\:6\log_{20}(3) \:<\:3

Divide by 6: . . . . . . . . . . . . . \frac{1}{3} \:<\:\log_{20}(3) \:<\:\frac{1}{2}



Top
 Profile  
 
 Post subject: Re: log inequality
PostPosted: Thu, 17 May 2012 07:52:53 UTC 
Offline
Moderator
User avatar

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6776
Location: On this day Taiwan becomes another Tiananmen under Dictator Ma.
jacks wrote:
How can I prove \displaystyle \frac{1}{3}<\log_{20}(3)<\frac{1}{2}


Follows immediately from 9<20<27.

_________________
\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


Top
 Profile  
 
 Post subject: Re: log inequality
PostPosted: Fri, 18 May 2012 16:43:46 UTC 
Offline
Member

Joined: Sun, 8 Apr 2012 09:36:59 UTC
Posts: 29
Thanks Soroban , Outermeasure


Top
 Profile  
 
Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 4 posts ] 

All times are UTC [ DST ]


Who is online

Users browsing this forum: No registered users


You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum

Search for:
Jump to:  
Contact Us | S.O.S. Mathematics Homepage
Privacy Statement | Search the "old" CyberBoard

users online during the last hour
Powered by phpBB © 2001, 2005-2011 phpBB Group.
Copyright © 1999-2013 MathMedics, LLC. All rights reserved.
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA