# What is meant by the "base" and "exponent" of a number and how do these relate to logarithms?

Mar 2, 2015

Exponentiation:

Base Definition:
( for integer exponent values greater than or equal to zero )

${a}^{x} = 1$ multiplied by $a$, $x$ times

example:
${3}^{5} = 1 \times 3 \times 3 \times 3 \times 3 \times 3$
(=243, if you're keeping track)

Note: based on this definition ${a}^{0} = 1$ for any value of $a$.

Important Insight:
${a}^{x} \times {a}^{y}$
= (1 xx a (x times)) xx (1 xx a (y times))
= (1 xx a (x+y times))#
$= {a}^{x + y}$

This insight is used to extend the definition of exponentiation to include negative and fractional exponents

${a}^{- x} = \frac{1}{{a}^{x}}$

${a}^{\frac{1}{x}} = {x}^{t h}$ root of $a$ (sorry; I don't know how to get the symbols for this)

Also
$\frac{{a}^{x}}{{a}^{y}} = {a}^{x - y}$
and
${\left({a}^{x}\right)}^{y} = {\left({a}^{y}\right)}^{x} = {a}^{x y}$

Logarithms:

$\log$ with some specified base is a function.

Definition:
The value of ${\log}_{b} \left(m\right) =$ the value of $y$ needed to make ${b}^{y} = m$
(Repeat that multiple times before going on. Come back to it as often as you need).

Example:
${\log}_{3} \left(81\right) = 4$ since ${3}^{4} = 81$

Common log identities:
${\log}_{b} \left(b\right) = 1$

${\log}_{b} \left({b}^{x}\right) = x$

${\log}_{b} \left({c}^{x}\right) = x {\log}_{b} \left(c\right)$

${\log}_{b} \left(m n\right) = {\log}_{b} \left(m\right) + {\log}_{b} \left(n\right)$

${\log}_{b} \left(\frac{m}{n}\right) = {\log}_{b} \left(m\right) - {\log}_{b} \left(n\right)$

${\log}_{b} \left({m}^{n}\right) = n {\log}_{b} \left(m\right)$

Some special $\log$ notes:

• Sometimes you may see $\log \left(m\right)$ written without a base specified; the convention in this case is that the base is $10$
• Often you will see $\ln \left(m\right)$; this is another form of the $\log$ function with a special base value, $e$; that is
$\ln \left(m\right) = {\log}_{e} \left(m\right)$
where $e$ is a special number (like $\pi$) approximately equal to $2.72$; $e$ has some special properties that make it useful in calculus.