Logarithm

Definition

A Function that returns the exponent you need to get a argument from a given base.

• Base: $0<x<1\cup x>1$
• Argument: $0<x$
• Exponential: $x\in R$

Log Properties

Only valid if b is positive ${\log }_b(1)=0$ ${\log }_b(b)=1$ ${\log }_b(b^x)=x$ $b^{{\log }_{b}x}=x$ $e^{\ln x}=x$ $lo{g}_{b}x=\frac{1}{{\log }_{x}b}$

• Logs argument are never negative
• Log bases are never negative unless you use Imaginary Number
• Log base can never be one

Theorems

Change of Base Theorem

$\frac{\log t}{\log b}={\log }_{b}t=x$

Power Law

${\log }_b(x^n)=n{\log }_{b}x$

Product Law of Logarithms

${\log }_b(mn)={\log }_{b}m+{\log }_{b}n$

Quotient Law of Logarithms

${\log }_b(\frac{m}{n})={\log }_b(m)-{\log }_b(n)$

Guides

• Graphing Logarithms
• Inverse Exponentials

Examples

• Example with Transformation

Concepts

• Natural Logarithms
• Iterated Logarithm