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

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

### 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 $0$.