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# Properties of Logarithmic and Exponential Functions

EXP1Definition 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.

EXP2Definition of the Natural Exponential Base: The exponential base "e" is equal to the irrational number 2.718281828459... and is also defined as

EXP3Horizontal 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.

EXP4Function 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.

EXP5Definition 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

EXP6Inverse 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

EXP7Log 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.

EXP8Log 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.

EXP9Product 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.

EXP10Quotient 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.

EXP11Power Rule for Logs: Given the logarithmic function  f(x) = logax, f(xN) = N●f(x).

In other words, loga(xN) = N● logax.

EXP12Change 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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