3.9 Exponential and logarithmic functions  (Page 2/3)

The nature of exponential function are different around a=1. The plots of exponential functions for two cases (i)0<a<1 and (ii) a>1 are discussed here. If the base is greater than zero, but less than “1”, then the exponential function asymptotes to positive x-axis. It is easy to visualize the nature of plot. It is placed in the positive upper part as "f(x)" is positive. Also, note that ${\left(0.25\right)}^2$ is greater than ( ${\left(0.25\right)}^4$ . Hence, plot begins from a higher value to lower value as "x" increases, but never becomes equal to zero.

If the base is greater than “1”, then the exponential function asymptotes to negative x-axis. Again, it is easy to visualize the nature of plot. It is placed in the positive upper part as "f(x)" is positive. Also, note that ${\left(1.25\right)}^2$ is less than ( ${\left(1.25\right)}^4$ . Hence, plot begins from a lower value to higher value as "x" increases.

Note that expanse of exponential function is along x – axis on either side of the y-axis, showing that its domain is R. On the other hand, the expanse of “y” is limited to positive side of y-axis, showing that its range is positive real number. Further, irrespective of base values, all plots intersect y-axis at the same point i.e. y = 1 as :

$$y=a^x=a^0=1$$

Logarithmic functions

A logarithmic function gives “exponent” of an expression in terms of a base, “a”, and a number, “x”. The following two representations, in this context, are equivalent :

$$a^y=x$$

and

$$f\left(x\right)=y={\log }_{a}x$$

where :

• The base “a” is positive real number, but excluding “1”. Symbolically, $a>\mathrm{0,}a\ne 1$ .
• The number “x” represents result of exponentiation, “ $a^y$ ” and is also a positive real number. Symbolically, x>0.
• The exponent “y” i.e. logarithm of “x” is a real number.

Note that neither “a” nor “x” equals to zero.

The expression of a logarithm for “x” on a certain base represents logarithmic function. In words, we can say that a logarithmic function associates every positive real number (x) to a real valued exponent (y), which is symbolically represented as :

$$f\left(x\right)=y=\log {}_{a}x;a,x>\mathrm{0,}a\ne 1$$

Following earlier discussion for the case of exponential function, we exclude "a = 1" as logarithmic function is not relevant to this base.

$$1^y=1$$

We can easily see here that whatever be the exponent, the value of logarithmic function is “1". Hence, base “1” is irrelevant as exponent “y” is not uniquely associated with “x”.

From the defining values of "x" and "f(x)", we conclude that domain and range of logarithmic function is :

$$\text{Value of “x”}=\text{Domain}=\left(\mathrm{0,}\infty \right)$$

$$\text{Value of “y”}=\text{Range}=R$$

Note that domain and range of logarithmic function is exchanged with respect to domain and range of exponential function.

Base

The base of the logarithmic function can be any positive number. However, “10” and “e” are two common bases that we often use. Here, “e” is a mathematical constant given by :

$$e=2.718281828$$

If we use “e” as the base, then the corresponding logarithmic function is called “natural” logarithmic function. The plots, here, show logarithmic functions for two bases (i) 10 and (ii) e.