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MATH 1441
Technical Mathematics for Biological Sciences
Basic Properties of Exponents and Exponential Functions
Basic Properties of Exponents
Here, we just remind you of a few very basic principles.
Although exponents started out as a shorthand for indicating repeated products (so that, for example, the
exponent '3' in x3 indicates a product consisting of three factors of x), the notation is easily and meaningfully
generalized to allow exponents which are not just positive whole numbers. We will assume you are able to
explain what is meant by xn when n is any positive real number.
Remember that when an exponent is used, it applies only to the quantity or symbol immediately to its lower
left. Thus:



in 5x2, the exponent 2 applies only to the x. To make it also apply to the 5, you need to employ
brackets: (5x)2 = (5x)(5x) = 52x2. This indicates that exponents propagate through a product.
this rule applies to negative numbers as well. Thus -52 = -25, since the exponent applies only
to the 5, not to the minus sign. To indicate the square of the negative number, -5, you would
have to use brackets again: (-5)2 = 25.
the only really useful arithmetic that can be done with exponentials is multiplication, division,
and raising to a power. The rules are:
am  an  am  n
am
an
 am  n
a 
m n
 a mn
Properties of Exponential Functions
It is useful to distinguish between algebraic powers such as x2 where the base is a variable and the power or
exponent is a constant, and exponential functions such as 2x, where the base is a constant and the
exponent is a variable. Because we usually picture the exponent in an exponential function as having an
real number value, we use the term "exponent" in preference to the term "power".
It is helpful to be able to visualize exponential functions,
and the difference between an exponential function and
an algebraic power function.
y
y = 2x
The figure to the left shows a plot of y = x2 and y = 2x on
the same set of axes. From this you can see that the
exponential function tends to increase much faster than
the algebraic power function for positive values of the
exponent, with the effect becoming more pronounced
the larger the value of x.
This figure also shows the typical shape of the graph of
an exponential function. For positive values of the
exponent, the function value increases rapidly. As you
move leftwards from x = 0, however, the value of the
exponential function decreases slowly towards the value
of zero (though it never reaches zero exactly). Note that
David W. Sabo (1999)
Basic Properties of Exponents and Exponential Functions
y = x2
x
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an exponential function can never have a negative value, nor can it ever be exactly equal to zero.
In the graph on the left, the functions y = 2x and
y = 2-x are compared. The function, y = 2x,
increases gradually from near zero (when x is large
and negative) to the value 1 (when x = 0) and then
increases rapidly as x increases to the right. This is
the characteristic shape of exponential growth. On
the other hand, the function y = 2-x starts with a large
positive value when x is large and negative,
decreases rapidly to the value 1 when x = 0, and
then continues to decrease gradually as x increases
to the right. This is the characteristic shape of
exponential decay.
y
y = 2-x
Finally, the last graph to the right contrasts the functions y = 2x
and y = 3x. As expected, the function with the larger base
increases more rapidly than the function with the smaller base
as x increases from zero. (However, to the left of x = 0, the two
graphs swap positions.) Since e  2.718, which is between 2
and 3, the graph of y = ex would be intermediate between the
two plots shown in the figure.
y = 2x
x
y
y = 3x
y = 2x
x
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Basic Properties of Exponents and Exponential Functions
David W. Sabo (1999)