The notion of exponential function

Exponential and logarithmic growth: Exponential growth

The notion of exponential function

Cell division in the formation of an embryo is an example of repetitive duplication. The number of cells in the embryo goes from \(1\) to \(2\) to \(4\) to \(8\), and so on. A formula for the number of cells in the embryo after a certain number of cell divisions \(x\) is therefore equal to \(2^x\).

This is an example of a multiplication by a fixed factor, in this case \(2\).

Exponential function

Let #g# be a positive real number. The function

\[f(x) = g^x\] is called the exponential function with base or growth factor \(g\).

The argument #x# of this function is called exponent. Indeed, it is the exponent of the exponentiation with base #g#.

The function value is exact at times, but an approximation can always be calculated:

\(4^{-2} = \frac{1}{4^2} = \frac{1}{16}\) is an exact calculation of the value of the exponential function #4^x# at #-2#.
\(1.3^\frac{1}{2} = \sqrt{1.3}\) is an exact calculation of the value of the exponential function #1.3^x# for #x=\frac{1}{2}#.
\(10^{5.7} = 10^5 \cdot 10^{0.7} \approx 5.01 \cdot 10^5\) is an approximation of the value of the exponential function with base #10# at #5.7#.

In the introduction, \(x\) is the number of cells in an embryo, and therefore always a positive integer. In general, \(x\) can assume all real values. In other words, the exponential function is defined on the whole real line.

Unlike a power function such as \(x^3\), the argument (variable #x#) of an exponential function appears in the exponent: \(3^x\). Hence the name exponential function.

Mathematicians often speak of #g# as the base of the exponential function #g^x#. In Financial Calculus, the name growth factor is more common.

The growth factor is the relative increase of the function value per unit increase of \(x\): each time \(x\) increases by \(1\), the function value is multiplied by the growth factor.

Monotony of the exponential function

The exponential function \(f(x) = g^x\) is decreasing if \(0\lt g\lt 1\) and increasing if \(g\gt 1\).

Suppose \(0\lt g\lt 1\). We use the fact that #g^a\gt1# for #a\lt0#. If #x# and #y# are real numbers with #x\lt y#, then #x-y# is negative so #g^{x-y}\gt 1#. Multiplying both sides by the positive number #g^y# we find #g^{x}\gt g^y#. In other words, if the value of #x# increases (like #y#), then the value of #g^x# decreases. This shows that the exponential function \(f(x) = g^x\) decreases if \(0\lt g\lt 1\).

Suppose \(g\gt 1\). We use the fact that #g^a\gt1# for #a\gt0#. If #x# and #y# are real numbers with #x\lt y#, then #y-x# is positive so #g^{y-x}\gt 1#. Multiplying both sides by the positive number #g^x# gives #g^{y}\gt g^x#. In other words, if the value of #x# increases (like #y#), then the value of #g^x# increases. This shows that the exponential function \(f(x) = g^x\) increases if \(g\gt 1\).

Each exponential function with \(g \gt0\) and \(g \neq 1\) has a horizontal asymptote, \(y = 0\). Here, this means that the function value never decreases below \(0\), and that it comes arbitrarily close to it (if you take #x# far enough away from #0# and negative in the case #g\gt1# and if you take #x# large enough in case #0\lt g\lt1#).

The case #g=1# is excluded; then #g^x# is equal to the constant function #1#. In other words, the function value is always equal to #1#.

For the exponential function with growth factor \(3\), we have the formula \(f(x) = 3^x\).

The table shows how rapidly the function value increases:

\(x\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)
\(f(x)\)	\(\frac{1}{3}\)	\(1\)	\(3\)	\(9\)	\(27\)	\(81\)

After each step of size #1# (that is, upon replacing #x# by #x+1#) the function value triples:

\[f(x+1) = 3^{x+1} =3\cdot 3^x = 3\cdot f(x) \]

If \(x\) increases by \(1\), the function values increases by \(2\cdot f(x)\).

You can also see rapid increases and decreases in the following graph \(g^x\): by use of the slider you can set the value of the growth factor \(g\). Check out what happens when \(g=1\) and when \(g\lt 1\).