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

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| style="background:lime; border:1px solid black;height:200px;padding:10px;" | {{User:Temperal/testtemplate|page 2}}
 
| style="background:lime; border:1px solid black;height:200px;padding:10px;" | {{User:Temperal/testtemplate|page 2}}
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==<span style="font-size:20px; color: blue;">Simple Number Theory</span>==
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==<span style="font-size:20px; color: blue;">Exponentials and Logarithms</span>==
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This is a collection of essential AIME-level number theory theorems and other tidbits.
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This is just a quick review of logarithms and exponents; it's elementary content.
 +
===Definitions===
 +
*Exponentials: Do you really need this one?
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*Logarithms: If <math>b^a=x</math>, <math>\log_b{x}=a</math>. Note that a logarithm in base [[e]], i.e. <math>\log_e{x}=a</math> is notated 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.
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===Rules of Exponentiation and Logarithms===
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<math>a^x \cdot a^y=a^{x+y}</math>
  
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===Trivial Inequality===
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<math>(a^x)^y=a^{xy}</math>
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For any real <math>x</math>, <math>x^2\ge 0</math>, with equality iff <math>x=0</math>.
 
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===Arithmetic Mean/Geometric Mean Inequality===
 
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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>\frac{a^x}{a^y}</math>=a^{x-y}<math>
  
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===Cauchy-Schwarz Inequality===
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</math>a^0=1<math>, where </math>a\ne 0<math>.
  
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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>\log_b{xy}=\log_b{x}+\log_b{y}<math>
  
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<math>(\sum a_i^2)(\sum b_i^2) \ge (\sum a_ib_i)^2</math>
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</math>\log_b{x^y}=y\cdot \log_b{x}<math>
  
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====Cauchy-Schwarz Variation====
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</math>\log_b{\frac{x}{y}}=\log_b{x}-\log_b{y}<math>
  
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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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</math>\log_b{a}=\frac{1}{\log_a{b}}<math>
  
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<math>\sum\left({{a_i^2}\over{b_i}}\right) \ge {{\sum a_i^2}\over{\sum b_i}}</math>.
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</math>\log_b{b}=1<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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[[User:Temperal/The Problem Solver's Resource1|Back to page 1]] | [[User:Temperal/The Problem Solver's Resource3|Continue to page 3]]
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</math>\log_1{a}<math> and </math>\log_0{a}$ are undefined.
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|}<br /><br />
 

Revision as of 16:24, 29 September 2007



The Problem Solver's Resource
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?
  • Logarithms: If $b^a=x$, $\log_b{x}=a$. Note that a logarithm in base e, i.e. $\log_e{x}=a$ is notated 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 and Logarithms

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

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

$\frac{a^x}{a^y}$=a^{x-y}$$ (Error compiling LaTeX. Unknown error_msg)a^0=1$, where$a\ne 0$.$\log_b{xy}=\log_b{x}+\log_b{y}$$ (Error compiling LaTeX. Unknown error_msg)\log_b{x^y}=y\cdot \log_b{x}$$ (Error compiling LaTeX. Unknown error_msg)\log_b{\frac{x}{y}}=\log_b{x}-\log_b{y}$$ (Error compiling LaTeX. Unknown error_msg)\log_b{a}=\frac{1}{\log_a{b}}$$ (Error compiling LaTeX. Unknown error_msg)\log_b{b}=1$$ (Error compiling LaTeX. Unknown error_msg)\log_b{a}=\frac{\log_x{a}}{\log_x{b}}$, where x is a constant.$\log_1{a}$and$\log_0{a}$ are undefined.

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