Difference between revisions of "User:Temperal/The Problem Solver's Resource2"
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==<span style="font-size:20px; color: blue;">Exponentials and Logarithms</span>== | ==<span style="font-size:20px; color: blue;">Exponentials and Logarithms</span>== | ||
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? If <math>a=b | + | *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=== |
<math>a^x \cdot a^y=a^{x+y}</math> | <math>a^x \cdot a^y=a^{x+y}</math> | ||
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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>. | ||
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+ | These should all be trivial and easily proven by the reader. | ||
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+ | ===Rules of Logarithms=== | ||
+ | <math>\log_b b=1</math> | ||
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+ | This can be seen by writing as <math>b^1=b</math>. | ||
<math>\log_b xy=\log_b x +\log_b y </math> | <math>\log_b xy=\log_b x +\log_b y </math> | ||
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<math>\log_b a=\frac{1}{\log_a b}</math> | <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. | <math>\log_b a=\frac{\log_x a}{\log_x b}</math>, 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. | |
+ | <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>. | ||
[[User:Temperal/The Problem Solver's Resource1|Back to page 1]] | [[User:Temperal/The Problem Solver's Resource3|Continue to page 3]] | [[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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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 .