Difference between revisions of "User:Temperal/The Problem Solver's Resource2"

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{{User:Temperal/testtemplate|page 2}}
{| style='background:lime;border-width: 5px;border-color: limegreen;border-style: outset;opacity: 0.8;width:840px;height:300px;position:relative;top:10px;'
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==<span style="font-size:20px; color: blue;">Exponentials and Logarithms</span>==
|+ <span style="background:aqua; border:1px solid black; opacity: 0.6;font-size:30px;position:relative;bottom:8px;border-width: 5px;border-color:blue;border-style: groove;position:absolute;top:50px;right:155px;width:820px;height:40px;padding:5px;">The Problem Solver's Resource</span>
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This is just a quick review of logarithms and exponents; it's elementary content.
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===Definitions===
| style="background:lime; border:1px solid black;height:200px;padding:10px;" | {{User:Temperal/testtemplate|page 2}}
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*Exponentials: Do you really need this one? If <math>a=\underbrace{b\times b\times b\times \cdots \times b}_{x\text{ }b'\text{s}}</math>, then <math>a=b^x</math>
==<span style="font-size:20px; color: blue;">Simple Number Theory</span>==
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*Logarithms: If <math>b^x=a</math>, then <math>\log_b{a}=x</math>. Note that a logarithm in base [[e]], i.e. <math>\log_e{x}=a</math> is denoted as <math>\ln{x}=a</math>, or the natural logarithm of x. If no base is specified, then a logarithm is assumed to be in base 10.
This is a collection of essential AIME-level number theory theorems and other tidbits.
 
  
===Trivial Inequality===
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===Rules of Exponentiation===
For any real <math>x</math>, <math>x^2\ge 0</math>, with equality iff <math>x=0</math><math>.
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<math>a^x \cdot a^y=a^{x+y}</math>
===Arithmetic Mean/Geometric Mean Inequality===
 
For any set of real numbers </math>S<math>, </math>\frac{S_1+S_2+S_3....+S_{k-1}+S_k}{k}\ge \sqrt[k]{S_1\cdot S_2 \cdot S_3....\cdot S_{k-1}\cdot S_k}<math> with equality iff </math>S_1=S_2=S_3...=S_{k-1}=S_k<math>.
 
  
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<math>(a^x)^y=a^{xy}</math>
  
===Cauchy-Schwarz inequality===
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<math>\frac{a^x}{a^y}=a^{x-y}</math>
  
For any real numbers </math>a_1,a_2,...,a_n<math> and </math>b_1,b_2,...,b_n<math>, the following holds:
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<math>a^0=1</math>, where <math>a\ne 0</math>.
  
</math>\displaystyle(\sum a_i^2)(\sum b_i^2) \ge (\sum a_ib_i)^2<math>
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These should all be trivial and easily proven by the reader.
  
====Cauchy-Schwarz variation====
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===Rules of Logarithms===
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<math>\log_b b=1</math>
  
For any real numbers </math>a_1,a_2,...,a_n<math> and positive real numbers </math>b_1,b_2,...,b_n<math>, the following holds:
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This can be seen by writing as <math>b^1=b</math>.
  
</math>\displaystyle\sum\left({{a_i^2}\over{b_i}}\right) \ge {{\sum a_i^2}\over{\sum b_i}}$.
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<math>\log_b xy=\log_b x +\log_b y </math>
  
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<math>\log_b x^y=y\cdot \log_b x </math>
  
[[User:Temperal/The Problem Solver's Resource1|Back to page 1]] | [[User:Temperal/The Problem Solver's Resource2|Continue to page 3]]
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<math>\log_b \frac{x}{y} =\log_b x-\log_b y</math>
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<math>\log_b a=\frac{1}{\log_a b}</math>
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<math>\log_b a=\frac{\log_x a}{\log_x b}</math>, where x is a constant.
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All of the above should be proven by the reader without too much difficulty - substitution and putting things in exponential form will help.
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<math>\log_1 a</math> and <math>\log_0 a</math> are undefined, as there is no <math>x</math> such that <math>1^x=a</math> except when <math>a=1</math> (in which case there are infinite <math>x</math>) and likewise with <math>0</math>.
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[[User:Temperal/The Problem Solver's Resource1|Back to page 1]] | [[User:Temperal/The Problem Solver's Resource3|Continue to page 3]]

Latest revision as of 18:22, 21 January 2016


Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 2.

Exponentials and Logarithms

This is just a quick review of logarithms and exponents; it's elementary content.

Definitions

  • Exponentials: Do you really need this one? If $a=\underbrace{b\times b\times b\times \cdots \times b}_{x\text{ }b'\text{s}}$, then $a=b^x$
  • Logarithms: If $b^x=a$, then $\log_b{a}=x$. Note that a logarithm in base e, i.e. $\log_e{x}=a$ is denoted as $\ln{x}=a$, or the natural logarithm of x. If no base is specified, then a logarithm is assumed to be in base 10.

Rules of Exponentiation

$a^x \cdot a^y=a^{x+y}$

$(a^x)^y=a^{xy}$

$\frac{a^x}{a^y}=a^{x-y}$

$a^0=1$, where $a\ne 0$.

These should all be trivial and easily proven by the reader.

Rules of Logarithms

$\log_b b=1$

This can be seen by writing as $b^1=b$.

$\log_b xy=\log_b x +\log_b y$

$\log_b x^y=y\cdot \log_b x$

$\log_b \frac{x}{y} =\log_b x-\log_b y$

$\log_b a=\frac{1}{\log_a b}$

$\log_b a=\frac{\log_x a}{\log_x b}$, where x is a constant.

All of the above should be proven by the reader without too much difficulty - substitution and putting things in exponential form will help.

$\log_1 a$ and $\log_0 a$ are undefined, as there is no $x$ such that $1^x=a$ except when $a=1$ (in which case there are infinite $x$) and likewise with $0$.

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