Utilization of Exponential Functions and their different properties
In this article, we will check out what an exponential function is. We can write the exponential function in the form: f(x) = bx, where b is any number but greater than zero and b is not equal to 1. Here, we refer to ‘b’ as the base and ‘x’ could be any real number.
In the above equation, we see that x is an exponent and the base number remains fixed. This is just the opposite of what we have figured out before. We have found out earlier that x represents a variable and the exponent represents a fixed number. No matter what, these functions get equated in the same way as it has been done before. Let us check the following using some examples.
Before getting into the concept, we should keep in mind the restrictions assigned for b. So we should avoid the numbers zero and one for that. If we use the restricted numbers, then the function looks like:
f(x) = bx = 0x = 0 and f(x) = bx = 1x = 1.
These functions do not have the properties of the exponential functions and are known as constant functions. Now, we should keep away from negative numbers so that we do not get any complex value while evaluating the function. Let us assume b as -2. Thus, according to the equation:
f(x) = bx = (-2)x
so, f(1/2) = (-2)1/2 = √-2.
Here we can see that a complex value is found while evaluating the exponential function. Thus we need to avoid negative numbers for b so that we get real numbers as values.
Properties of the exponential function
- When plotting over a graph, f(x) would always have the point (0, 1). We may also say that f(0) = 1, whatever be the value of ‘b’ be.
- For different values of b, we can say bx > 0 and bx ≠ 0.
- If b is greater than zero but less than one, then the graphical representation of bx will decrease from left to right.
- Similarly, if b is greater than one, then the graphical representation of b will increase from left to right.
- Another equation can also be formed by using bx = by. Then, we can say that x is equal to y.
Another major thing to point out in this section is the special exponential function. This is exactly what the original exponential function looks like: f(x) = ex. We can check out the difference between the two functions, f(x) = bx and f(x) = ex. In the first function, b can be any real number greater than zero but not equal to one whereas in the second function, e is a specific number. Here the value of e is equal to 2.718281828. This special exponential function is of paramount importance and is confronted in many areas.
Rules of exponential functions:
- Product Rule: To find out the product of two exponents in the same base, we need to add the exponents. For example: 4x*45x = 46x.
- Quotient Rule: To find out the quotient of two exponents with the same base, we need to subtract the exponents. For example: 44x/42x = 42x.
- Power Rule: When an exponent is taken to a power, then we multiply the two known powers. For example: (63x)3 = 69x.
- Zero Rule: Any number with a power of zero is always one. For example: a0 = 1.
- Negative Rule: If an exponent is taken as a negative number, we can easily utilize them as positive by placing them in the denominator. For example: 2-2x = 1/22x.
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