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

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This is just a quick review of logarithms and exponents; it's elementary content.
 
This is just a quick review of logarithms and exponents; it's elementary content.
 
===Definitions===
 
===Definitions===
*Exponentials: Do you really need this one?
+
*Exponentials: Do you really need this one?  
 
*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.
 
*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.
 
===Rules of Exponentiation and Logarithms===
 
===Rules of Exponentiation and Logarithms===
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</math>a^0=1<math>, where </math>a\ne 0<math>.
 
</math>a^0=1<math>, where </math>a\ne 0<math>.
  
</math>\log_b{xy}=\log_b{x}+\log_b{y}<math>
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</math>\log_b xy=\log_b x +\log_b y <math>
  
</math>\log_b{x^y}=y\cdot \log_b{x}<math>
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</math>\log_b x^y=y\cdot \log_b x <math>
  
</math>\log_b{\frac{x}{y}}=\log_b{x}-\log_b{y}<math>
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</math>\log_b \frac{x}{y} =\log_b x-\log_b y<math>
  
</math>\log_b{a}=\frac{1}{\log_a{b}}<math>
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</math>\log_b a=\frac{1}{\log_a b}<math>
  
</math>\log_b{b}=1<math>
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</math>\log_b b=1<math>
  
</math>\log_b{a}=\frac{\log_x{a}}{\log_x{b}}<math>, where x is a constant.
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</math>\log_b a=\frac{\log_x a}{\log_x b}<math>, where x is a constant.
  
</math>\log_1{a}<math> and </math>\log_0{a}$ are undefined.
+
</math>\log_1 a<math> and </math>\log_0 a$ are undefined.

Revision as of 17:27, 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}$$ (Error compiling LaTeX. ! Missing $ inserted.)a^0=1$, where$a\ne 0$.$\log_b xy=\log_b x +\log_b y $$ (Error compiling LaTeX. ! Missing $ inserted.)\log_b x^y=y\cdot \log_b x $$ (Error compiling LaTeX. ! Missing $ inserted.)\log_b \frac{x}{y} =\log_b x-\log_b y$$ (Error compiling LaTeX. ! Missing $ inserted.)\log_b a=\frac{1}{\log_a b}$$ (Error compiling LaTeX. ! Missing $ inserted.)\log_b b=1$$ (Error compiling LaTeX. ! Missing $ inserted.)\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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