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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?  
*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^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 and Logarithms===
 
===Rules of Exponentiation===
<math>a^x \cdot a^y=a^{x+y}</math>
<math>a^x \cdot a^y=a^{x+y}</math>


<math>(a^x)^y=a^{xy}</math>
<math>(a^x)^y=a^{xy}</math>


<math>\frac{a^x}{a^y}</math>=a^{x-y}<math>
<math>\frac{a^x}{a^y}=a^{x-y}</math>
 
<math>a^0=1</math>, where <math>a\ne 0</math>.
 
These should all be trivial and easily proven by the reader.
 
===Rules of Logarithms===
<math>\log_b b=1</math>
 
This can be seen by writing as <math>b^1=b</math>.


</math>a^0=1<math>, where </math>a\ne 0<math>.
<math>\log_b xy=\log_b x +\log_b y </math>


</math>\log_b xy=\log_b x +\log_b y <math>
<math>\log_b x^y=y\cdot \log_b x </math>


</math>\log_b x^y=y\cdot \log_b x <math>
<math>\log_b \frac{x}{y} =\log_b x-\log_b y</math>


</math>\log_b \frac{x}{y} =\log_b x-\log_b y<math>
<math>\log_b a=\frac{1}{\log_a b}</math>


</math>\log_b a=\frac{1}{\log_a b}<math>
<math>\log_b a=\frac{\log_x a}{\log_x b}</math>, where x is a constant.


</math>\log_b b=1<math>
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_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</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>.


</math>\log_1 a<math> and </math>\log_0 a$ are undefined.
[[User:Temperal/The Problem Solver's Resource1|Back to page 1]] | [[User:Temperal/The Problem Solver's Resource3|Continue to page 3]]

Latest revision as of 09:58, 19 June 2025


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 $a=\underbrace{b\times b\times b\times \cdots \times b}_{x\text{ }b'\text{s}}$, then $a=b^x$
  • Logarithms: If $b^x=a$, then $\log_b{a}=x$. 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.

Rules of Exponentiation

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

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

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

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

These should all be trivial and easily proven by the reader.

Rules of Logarithms

$\log_b b=1$

This can be seen by writing as $b^1=b$.

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

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

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

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

$\log_b a=\frac{\log_x a}{\log_x b}$, 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.

$\log_1 a$ and $\log_0 a$ are undefined, as there is no $x$ such that $1^x=a$ except when $a=1$ (in which case there are infinite $x$) and likewise with $0$.

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