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===Definitions=== | ===Definitions=== | ||
*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> | *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> | ||
− | *Logarithms: If <math>b^a | + | *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. |
===Rules of Exponentiation=== | ===Rules of Exponentiation=== |
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 , then
- Logarithms: If , then . Note that a logarithm in base e, i.e. is denoted as , or the natural logarithm of x. If no base is specified, then a logarithm is assumed to be in base 10.
Rules of Exponentiation
, where .
These should all be trivial and easily proven by the reader.
Rules of Logarithms
This can be seen by writing as .
, 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.
and are undefined, as there is no such that except when (in which case there are infinite ) and likewise with .