In mathematics, we might have come across different types of functions such as polynomial functions, even functions, odd functions, rational functions and trigonometric functions, etc. In this article, you will learn about a new classification of functions called exponential functions and logarithmic functions.

What are Exponential and Logarithmic Functions?

Exponential Function Definition:

An exponential function is a Mathematical function in the form y = f(x) = bx, where “x” is a variable and “b” is a constant which is called the base of the function such that b > 1. The most commonly used exponential function base is the transcendental number e, and the value of e is equal to 2.71828.

Using the base as “e” we can represent the exponential function as y = ex. This is called the natural exponential function. However, an exponential function with base 10 is called the common exponential function.

Learn more about exponential functions here.

Logarithmic Function Definition:

If the inverse of the exponential function exists then we can represent the logarithmic function as given below:

Suppose b > 1 is a real number such that the logarithm of a to base b is x if bx = a.

The logarithm of a to base b can be written as logb a.

Thus, logb a = x if bx = a.

In other words, mathematically, by making a base b > 1, we may recognise logarithm as a function from positive real numbers to all real numbers. This function is known as the logarithmic function and is defined by:

logb : R+ → R

x → logb x = y if by = x

If the base b = 10, then it is called a common logarithm and if b = e, then it is called the natural logarithm. Generally, the natural logarithm is denoted by ln.

Read more:

Properties of Exponential and Logarithmic Functions

Some of the prominent features of the exponential functions are listed below:

  • The domain of the exponential function is the set of all real numbers, i.e. R.
  • The range of the exponential function is the set of all positive real numbers.
  • The point (0, 1) is always on the graph of the given exponential function since it supports the fact that b0 = 1 for any real number b > 1.
  • The exponential function is ever increasing; i.e., as we move from left to right, the graph rises above.
  • For the large set of negative values of x, the exponential function is very close to 0; for example, the graph approaches the x-axis but never meets it.

These can be observed from the graph of an exponential function given below:

Exponential and logarithmic functions 1

Some of the essential considerations on the logarithm function to any base b > 1 are listed below:

  • It is not possible to derive a meaningful definition of the logarithm for non-positive numbers, i.e. for negative numbers. So, the domain of the log function is the set of positive real numbers, i.e. R+.
  • The range of the log function is the set of all real numbers.
  • The point (1, 0) is always on the graph of the log function.
  • The log function is ever-increasing, i.e., as we move from left to right the graph rises above.
  • For the value of x quite near to zero, the value of log x can be made lesser than any given real number. That means, in quadrant IV, the graph approaches the y-axis but never meets it.

Exponential and Logarithmic functions 2

In the above graph of y = ex and y = ln x, we observe that the two curves are the mirror images of each other reflected over the line y = x.

Rules of Exponential and Logarithmic Functions

Below are the rules of exponential functions and logarithmic functions.

Exponential Rules Logarithmic Rules
  • ax ay = ax+y
  • ax/ay = ax-y
  • (ax)y = axy
  • Axbx = (ab)x
  • (a/b)x = ax/bx
  • a0 = 1
  • a-x = 1/ ax
  • logb (xy) = logb x + logb y
  • logb (x/y) = logb x – logb y
  • logb xm = m logb x
  • logb p2 = logb p + logb p = 2 log p
  • loga p = (logb p)/ (logb a)
  • logb 1 = 0
  • logb b = 1
  • logb bx = x

Exponential and Logarithmic Functions derivatives

The derivatives of exponential and logarithmic functions formulas are given below.

The derivative of ex with respect to x is written as:

\(\begin{array}{l}\frac{d}{dx}(e^{x})=e^x\end{array} \)

The derivative of log x with respect to x is written as:

\(\begin{array}{l}\frac{d}{dx}(log\ x)=\frac{1}{x}\end{array} \)

Using these two formulas, we can derive other formulas by applying exponential and logarithmic rules.

  • The derivative of eax with respect to x is:
    \(\begin{array}{l}\frac{d}{dx}(e^{ax})=ae^{ax}\end{array} \)
  • The nth derivative of eax with respect to x is:
    \(\begin{array}{l}\frac{d^n}{dx}(e^{ax})=a^ne^{ax}\end{array} \)

Similarly, we can derive multiple formulas for the derivative of exponential functions.

Exponential and Logarithmic Functions Examples

Example 1:

Find the derivative of log (log x), x > 1, with respect to x.

Solution:

\(\begin{array}{l}\frac{d}{dx}y = \frac{d}{dx}log(log\ x)\end{array} \)

Using the formula d/dx (log x) = 1/x,

\(\begin{array}{l}=\frac{1}{log\ x}\ \frac{d}{dx}(log\ x)\end{array} \)
\(\begin{array}{l}=\frac{1}{log\ x}. \frac{1}{x}\end{array} \)

Therefore, (d/dx) [log (log x)] = 1/(x log x).

Example 2:

Simplify: y = 135/133

Solution:

y = 135/133

Using the quotient rule,

y = 135/133 = 13(5-3)

= 132

= 169

Example 3:

Evaluate the derivative of the function

\(\begin{array}{l}y=e^{x^4}\end{array} \)
.

Solution:

Given function is:

\(\begin{array}{l}y=e^{x^4}\end{array} \)

Differentiating with respect to x,

\(\begin{array}{l}\frac{dy}{dx}=\frac{d}{dx}(e^{x^4}) = e^{x^4}\frac{d}{dx}(x^4)=e^{x^4}.4x^{4-1}=4x^3e^{x^4}\end{array} \)

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