While reading newspapers or while watching the population growth rate you might have come across the word exponential. If something is happening rapidly we use the term exponential for that. Ex: The price of petrol has increased at an exponential rate. In mathematics, Exponentiation is written as , where b is the base and n is the exponent. Functions involving exponents are calledexponential functions.
In mathematics, an exponential function is a function of the form f (x) = bx, where “x” is a variable and “b” is a constant which is called the base of the function and it should be greater than 0. Ex: . If the base of the function is equal to 1 then the function f(x) = 1 this represents a linear function but not the exponential function. Hence the value of the base should be greater than 1 for an exponential function. In a math class if a student asks you about an exponential function. You can explain it with an example of a bacterial growth rate. The most popular exponential function also known as the natural exponential function is f(x) = , where ‘ is “Euler’s number”.
1. Growth of a quantity initially increases slowly and then suddenly increases.
2. The graph of exponential growth is increasing.
3. The formulas of exponential growth are used in calculating compound interest, cell division rate, etc.
4. Examples of exponential growth are f(x) = pbx, where b > 1, L = L0 where b = 1+r
1. Growth of a quantity initially decreases rapidly and then decreases slowly.
2. The graph of exponential growth is decreasing.
3. The formulas of exponential growth are used in modeling pandemic disease control rate, stock rate decrease, etc.
4. Examples of exponential growth are f(x) = pbx, where 0 > b > 1, L = L0 where b = 1-r .
The power series defines the real exponential function
ex = = (1/1) + (x/1) + (x2/2) + (x3/6) + …
Some other exponential functions are expanded as shown below,
= = (1/1) + (1/1) + (1/2) + (1/6) + …
= = (1/1) – (1/1) + (1/2) – (1/6) + …
The rules of exponential function are the same as those of exponents. They are
1. Law of Zero Exponents: p0 = 1
2. Law of Product: pm × pn = pm+n
3. Law of Quotient: pm/pn = pm-n
4. Law of Power of a Power: (pm)n = pmn
5. Law of Power of a Product: (pq)m = pmqm
6. Law of Power of a Quotient: (p/q)m = pm/qm
7. Law of Negative Exponents: p-m = 1/pm
Inverse of exponential function can be written by using a logarithmic function.
px is the inverse function of logpx.
Simplify the exponential equation 1. 3x+1 3x 2. f(x) = (¼)x for x = 1/2
Solution: 1. 3x+1 3x
By the rules of exponential functions, we have Law of Quotient: pm/pn = pm-n
3x+1 3x = 3 x+1- x = 3
2. f(x) = (¼)x for x = ½
Solution: f(½) = (¼) ½ = = ½
With this, the concept of exponential functions is ended. If you still want to learn more about it then log on to the Cuemath website. It will give complete detailed information about the topic. You can also attend the math classes of Cuemath. Well-trained teachers will make you understand the concept in a much better way.
Internal link – Opticalworlds