Exponential Function Calculator

Exponential Function Calculator

Exponential function calculator finds the values of the unknown parameters in the exponential function and provides the solution of the exponential function value that is based on some parameters. 

What is an Exponential Function?

Exponential function is a mathematical function that characterizes the relationship between input and output. It is used in the repeated multiplication of any initial value to find the output of any given input. 

An exponential function in the mathematical form “f(x) = ab^x” where “x” is a variable or known as an exponent, “a” is the constant, and “b” is a positive constant or known as base.  In simple form can be stated as “f(x) = b^x”.

The most commonly used base is Euler’s number “e” (approx.2.71828), which leads to the natural exponential function “f(x) = e^x”. This type of function is notable for rapid growth or decay. It is commonly used in various practical fields of science such as population growth and radioactive decay. 

Exponential Function Formula

Exponential function formulas are typically written in several general forms. These are used in various applications of mathematics for different real-world scenarios. Formulas can be stated as below:

1. f(x) = b^x
2. f(x) = ab^x
3. f(x)= a⋅ b^c⋅x+p +q

In these functions, the value of “b” shows the behaviors of function is increases or decreases: 

• If b >1, the function exhibits exponential growth or increases behavior.
• If 0 < b < 1, it exhibits exponential decay or decrease behavior.

Some other formulas of exponential function in the form of base “e”:

1. f(x) = e^cx
2. f(x) = a.e^cx

These function exponential growth or decay depending on the value of “c”:

• If “c>0 & x>0” then f(x) shows exponential growth.
• If “c<0 & x>0” then f(x) represents exponential decay.

Our above exponential function calculator helps to find the unknown parameter of all the above formulas. Anyone can quickly find results and understand the behavior of different exponential functions easily.

How to Calculate Exponential Function?

In this section, we’ll perform some examples of exponential functions, in which we calculate the unknown parameters by using the value of the function and one known parameter. Also, demonstrates the methods to find the exponential functions using a point’s value and the value of the parameter.
Example 1: Solve the exponential function and find the base “b” when “f (x) = 4 & x = 2”. The given function is defined as “f(x) = b^x”.

Solution:

Step 1: Substitute the values in the given function.

f(x) = b^x

4 = b²

Step 2: Solve the above expression for “b” by applying the square root of both sides.

√4 =√ b²

 b = 2

Thus, the base “b = 2” and the function becomes “f(x) = 2^x”.

Example 2: Find the unknown parameter of “a =?, b=?” for the given function “f(x) = a.b^x”. If the two points and function values are given as: x1 = 2, f(x1) = 6, x2 = 4, f(x2) = 9. 

Solution:

Step 1: Substitute the values in the given form of the function for both points.

f(x1) = a.b^x1

6 = a.b²   …… (i)

f(x₂) = a.b^x2

9 = a.b⁴    …… (ii)

Step 2: Now, divide the equation (i) by (ii) to find the constant values.      

6/9 = a.b²/ a.b⁴

6/9 = b²/ b⁴

2/3 = 1/b^(4-2)

Arrange the factor to find the unknown value.

b² = 3/2

Taking square roots on both sides.

√b² = √3/2

b = 1.22

Step 3: Now, put the value of “b” in eq (i).

6 = a.b²

6 = a.(1.22)²

6 = a. (1.4884)

a = 6/(1.4884)

a = 4.03

Thus, the constant “a = 4.03, b = 1.22” and the function becomes “f(x) = (4.03).(1.22)^x”.

Example 3: Evaluate the given function value “f(x) = a. b^(cx+p) + q” if the values are given as:

a = 4,    b =2,    c = 3,      p = 6,       q = 5,    x = 2   

Solution:

Step 1: Substitute the values in the given function.

f(x) = a. b^(cx+p) + q

f(2) = 4. 2^(3.2+6) + 5 ... (i)

Step 2: Now, first simplify the exponent part.

2^(3.2+6) = 2⁽⁶⁺⁶⁾= 2⁽¹²⁾ = 4096

Step 3: Substitute the exponent into the equation (i) and simplify.

f(2) = 4. (4096) + 5

=16384 + 5

=16389

Thus, the value of the function for the given values is “16389”.  

To verify the solution of all the above examples use exponential function calculator, which provides the unknown parameter and function values accurately.