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) = ax,
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) =
ax, 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) = ax,
The graph of g(x) = a-x will be a reflection of f(x)
across the y=axis.
The graph of h(x) = k●ax will be a vertical stretch of
f(x) but will have the same general shape.
The graph of p(x) = -k●ax will be a reflection of k●ax
across the x-axis.
EXP5 - Definition of The Logarithmic Function:
Given the exponential function y = ax, the equivalent
logarithmic function form is logay = x.
In other words, you may always rewrite logay = x
as y = ax and
you may always rewrite y = ax as logay = x
EXP6 - Inverse Property of The Logarithmic Function:
Given the exponential function f(x) = ax,
the inverse of f(x) is the logarithmic function form is f -1(x)
= logax.
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) = logax, f(1) = 0.
In other words, loga1 = 0 for any legitimate
exponential base a.
EXP8 - Log Property - Loga of a is 1:
Given the logarithmic function f(x) = logax, f(a) =
1.
In other words, logaa = 1 for any legitimate
exponential base a.
EXP9 - Product Rule for Logs: Given the
logarithmic function f(x) = logax, f(UV) = f(U) + f(V).
In other words, loga(UV) = logaU
+ logaV for any legitimate exponential base a.
EXP10 - Quotient Rule for Logs: Given the
logarithmic function f(x) = logax, 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) = logax, f(xN) =
N●f(x).
In other words, loga(xN) = N● logax.
EXP12 - Change of Base Rule for Logs: Given the
logarithmic function f(x) = logax, 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 base-e or 10), and rewrite our log term as a ratio of
two base-b logs.
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