3.9 Exponential and logarithmic functions

Logarithmic and exponential functions are closely related functions. Logarithmic functions are useful in interpreting expressions/ equations, in which exponents are unknown. On the other hand, exponential functions are representation of natural process or mathematical relations, having exponential growth or decay. We shall encounter many expressions, involving these two functions in mathematics, while analyzing or describing processes. Further, two function types have simple derivatives and related integration results (we shall learn about derivatives and integration in calculus).

Symbolically, exponential and logarithmic functions, respectively, are :

$$f\left(x\right)=a^x$$

and

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

In this module, we shall find out that exponential and logarithmic functions are inverse of each other. The roles of function "f(x)" and independent variable "x" are exchanged in two function definition. We shall clarify this point subsequently.

Exponential function

An exponential function relates every real number “x” to the exponentiation, “ $a^x$ ”. In other words, we can say that an exponential function associates every real number (x) to a function given by :

$$f\left(x\right)=y=a^x$$

where :

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

We need not memorize nature of "x","y" and "a". Rather we can investigate the same by reasoning. The base "a" can not be zero, because $0^x$ is not uniquely defined as required for invertible function. The exponential function is designed to be invertible. This means that "f(x)" and "x" be uniquely related. Also, if "a" is a negative number like "-2", then ${\left(-2\right)}^x$ may evaluate to positive or negative value depending on whether "x" is even or odd integer. Clearly sign of function depends on the nature of integer values of "x". In case, "x" is not an integer, then sign of ${\left(-2\right)}^x$ can not be interpreted for all values of "x". Further if a=1, then $1^x$ evaluates to "1" for all values of "x". Again function is not uniquely defined. In the nutshell, we conclude that base "a" is a positive number, but not equal to "1". In particular, the exponential function corresponding to base “e”, which is equal to 2.718281828, is called “natural” exponential function.

Once nature of "a" is decided, it is easy to find the nature of "y". Consider simplified exponents like ${10}^2$ , ${10}^1$ , ${10}^{-2}$ , ${10}^{-100}$ . All these numbers are greater than zero. This is true even if exponent is not integer. We conclude that "y" is positive number. Note that neither "a" nor "y" are zero. On the other hand, "x" by definition is a real number.

Clearly, the domain and range of exponential function are :

$$\text{Value of “x”}=\text{Domain}=R$$

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

We shall see subsequently that roles of “x” and “f(x)” are exchanged for logarithmic function. Here, “x” is the exponent i.e. the logarithmic value of a positive number and “f(x)” is the result of exponentiation, which is argument of logarithmic function. For this reason, we say that exponential and logarithmic functions are inverse to each other. As expected for inverse functions, we shall also see that domain and range of exponential and logarithmic functions are exchanged.