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

Revision as of 20:50, 10 January 2009

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

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

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 $b^a=x$ , $\log_b{x}=a$ . 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.

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

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

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

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

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

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

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

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

$\log_b b=1$

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

$\log_1 a$ and $\log_0 a$ are undefined.

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 This is just a quick review of logarithms and exponents; it's elementary content.
 ===Definitions===
-*Exponentials: Do you really need this one? If <math>a=\underbrace{b*b*b*...*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=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 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 and Logarithms===