Properties of Logarithmic and Exponential Functions
EXP1  Definition of an Exponential Function:
If f(x) is a function that may be written in the form f(x) = a^{x},
where a>0, a≠ 1, and x is any real number,
then f(x) is an exponential function and "a" is called the "base".
Note that functions that are compositions of an exponential
function and some other function are also often referred to as
exponential functions. For example, f(x) = 2●3^{2x}  1
would be referred to as a rational function.
EXP2  Definition of the Natural Exponential Base:
The exponential base "e" is equal to the irrational number
2.718281828459... and is also defined as
EXP3  Horizontal Asymptote of the Graph of a Basic
Exponential Function Graphs: Given the exponential function f(x) =
a^{x}, it's graph will have a horizontal asymptote of y=0 for
the left side only.
EXP4  Function Shift Rules Applied To Exponential
Function Graphs: Given the exponential function f(x) = a^{x},
The graph of g(x) = a^{x} will be a reflection of f(x)
across the y=axis.
The graph of h(x) = k●a^{x} will be a vertical stretch of
f(x) but will have the same general shape.
The graph of p(x) = k●a^{x} will be a reflection of k●a^{x}
across the xaxis.
EXP5  Definition of The Logarithmic Function:
Given the exponential function y = a^{x}, the equivalent
logarithmic function form is log_{a}y = x.
In other words, you may always rewrite log_{a}y = x
as y = a^{x} and
you may always rewrite y = a^{x} as log_{a}y = x
EXP6  Inverse Property of The Logarithmic Function:
Given the exponential function f(x) = a^{x},
the inverse of f(x) is the logarithmic function form is f^{ 1}(x)
= log_{a}x.
Also, since (f o f^{ 1})(x) = x and (f^{
1} o f)(x) = x
EXP7  Log Property  Log of 1 is 0: Given the
logarithmic function f(x) = log_{a}x, f(1) = 0.
In other words, log_{a}1 = 0 for any legitimate
exponential base a.
EXP8  Log Property  Log_{a} of a is 1:
Given the logarithmic function f(x) = log_{a}x, f(a) =
1.
In other words, log_{a}a = 1 for any legitimate
exponential base a.
EXP9  Product Rule for Logs: Given the
logarithmic function f(x) = log_{a}x, f(UV) = f(U) + f(V).
In other words, log_{a}(UV) = log_{a}U
+ log_{a}V for any legitimate exponential base a.
EXP10  Quotient Rule for Logs: Given the
logarithmic function f(x) = log_{a}x, f(U/V) = f(U) 
f(V).
In other words,
for any legitimate exponential base a.
EXP11  Power Rule for Logs: Given the
logarithmic function f(x) = log_{a}x, f(x^{N}) =
N●f(x).
In other words, log_{a}(x^{N}) = N● log_{a}x.
EXP12  Change of Base Rule for Logs: Given the
logarithmic function f(x) = log_{a}x, it is true, for
any legitimate bases a and b, that
In other words, we can pick whatever convenient base b that
we want (like basee or 10), and rewrite our log term as a ratio of
two baseb logs.
