The following figure represents the graph of exponents of x. It can be seen that as the exponent increases, the curves get steeper and the rate of growth increases respectively. Thus for x > 1, the value of y = fn(x) increases for increasing values of (n).
From the above discussion, it can be seen that the nature of polynomial functions is dependent on its degree. Higher the degree of any polynomial function, then higher is its growth. A function which grows faster than a polynomial function is y = f(x) = ax, where a>1. Thus, for any of the positive integers n the function f (x) is said to grow faster than that of fn(x).
Thus, the exponential function having base greater than 1, i.e., a > 1 is defined as y = f(x) = ax. The domain of exponential function will be the set of entire real numbers R and the range is said to be the set of all the positive real numbers.
It must be noted that exponential function is increasing and the point (0, 1) always lies on graph of exponential function. Also, it is very close to zero if the value of x is largely negative.
Exponential function having base 10 is known as common exponential function. Consider the following series:
The value of this series lies between 2 &3. it is represented by e. Keeping e as base the function, we get y = ex, which is a very important function in mathematics known as natural exponential function.
For a > 1, the logarithm of b to base a is x if ax = b. Thus, loga b = x if ax = b. This function is known as logarithmic function.
For base a = 10, this function is known as common logarithm and for base a = e, it is known as natural logarithm denoted by ln x. Following are some of the important observations regarding logarithmic functions which has base a>1.
- The domain of log function consists of positive real numbers only, as we cannot interpret the meaning of log functions for negative values.
- For the log function though the domain is only the set of positive real numbers, but the range is set of all real values, I.e. R
- When we plot the graph of log functions and move from left to right, the functions shows increasing behavior.
- The graph of log function never cuts x-axis or y-axis, though it seems to tend towards them.
- Logap = α, logbp = β and logba = µ, then aα = p, bβ = p and bµ = a
- Logbpq = Logbp + Logbq
- Logbpy = ylogbp
- Logb (p/q) = logbp – logbq
Let us now focus on the derivative of exponential and logarithmic functions.
- The derivative of exr.t. x is ex, I.e. d(ex)/dx = ex
- The derivative of log x w.r.t. x is 1/x, I.e. d(logx)/dx = 1/x