Exponential Function Calculator

Get the rapid and accurate evaluation of the used form of an exponential function with any parameter by the exponential function calculator with two points.

What Is The Exponential Function?

The type of function that involve exponents and their value is a constant raised to the power of the argument, especially the function where the constant is e.

In the exponential equation from two points calculator calculations, e is a mathematical constant that is equal to 2.71828. So, the basic exponent function is $$ f(t) = A0e^{kt} $$.

Types of Exponential Function:

There are two types of this function that are as follows:

Exponential growth
Exponential decay

When the quantity is increasing over time then it is referred to as exponential growth and when the quantity is decreasing over time then it is known as exponential decay.

How To Calculate The Exponential Function?

Now we calculate the exponent values of two functions with the help of an exponential function calculator. To clarify the concept look at the example.

Example:

The exponential calculator finds the function from two given data points (T1, Y1) and (T2, Y2). Suppose two functions (T1, Y1) and (T2, Y2) with the values (3, 2) and (4, 5) respectively. Calculate their time behavior at 4.

Given Data:

Time (T1) = 3
Function at T1 (Y1) = 2
Time (T2) = 4
Function at T2 (Y2) = 5

Solution:

The Generic form of an exponential function is:

$$ f(t) = A_0e^{kt} $$

To evaluate A 0 & k, we need to solve the following equations:

$$ y_1 = A_0e^{kt_1} $$

$$ y_2 = A_0e^{kt_2} $$

Step # 1:

Divide y1 by y2 to cancel A0

$$ \dfrac{y_1}{y_2} = \dfrac{A_0e^{kt_1}}{A_0e^{kt_2}} $$

$$ => \dfrac{y_1}{y_2} = \dfrac{\require{cancel}\cancel{A_0}e^{kt_1}}{\require{cancel}\cancel{A_0}e^{kt_2}} $$

$$ => \dfrac{y_1}{y_2} = \dfrac{e^{kt1}}{e^{kt_2}} $$

Step # 2:

Solving for k:

$$ \dfrac{y_1}{y_2} = \dfrac{e^{kt_1}}{e^{kt_2}} $$

$$ \dfrac{y_1}{y_2} = e^{kt_1}.e^{kt_2} $$

$$ \dfrac{y_1}{y_2} = e^{k(t_1 - t_2)} $$

Taking natural logs on both sides:

$$ In ({\dfrac{y_1}{y_2}}) = In(e^{k(t_1 - t_2)}) $$

$$ In ({\dfrac{y_1}{y_2}}) = e.k(t_1 - t_2) $$

$$ k = \dfrac{1}{t_1 - t_2} In ({\dfrac{y_1}{y_2}}) $$

Step # 3:

Now from 1 we have:

$$ y_1 = A_0e^{kt_1} $$ OR $$ A_0 = y_1e^{-kt_1} $$

Now put the value of k in the above equation:

$$ A_0 = y_1e^{-({\dfrac{1}{t_1 - t_2} In ({\dfrac{y_1}{y_2}})})t_1} $$

$$ A_0 = \require{cancel}\cancel{y_1} × \dfrac{y_2}{\require{cancel}\cancel{y_1}e^{kt_2}} $$

$$ A_0 = y_2e^{-kt_2} $$

Step # 4:

As we have calculated the formulas of k & A0, putting these values to calculate the actual values of k & A0.

$$ k = \dfrac{1}{3 - 4} In ({\dfrac{2}{5}}) $$

$$ k = 0.9163 $$

Now we have:

$$ A_0 = y_2e^{-kt_2} $$

$$ A_0 = 5×e^{-0.9163×4} $$

Step # 5:

Results:

$$ f(t) = A_0e^{kt} $$

$$ f(t) = 0.128e^{0.9163t} $$
Step # 6:
As the time at which we need to analyze the exponential function behavior is 2, so we have:

$$ f(2) = 0.128e^{0.9163×2} $$

$$ f(2) = 0.8 $$

You can also verify the answer by our exponential function from two points calculator. It also clarifies your concepts by showing the graph.

Exponential Function Graph:

exponential function calculator

Working of Exponential Function Calculator:

Our convert to exponential function formula calculator demands the following points to carry the final results of exponent functions.

Input:

Put the numeric expression T1 and their function value Y1
Enter the numeric expression T2 and their function value Y2
Set the point to evaluate that is optional
Tap “Calculate”

Output:

Our online exponential function given two points calculator will give you the following results.

The Exponential Function value
Graphical representation of an exponential function
It indicates either k is positive or negative
Step-by-step calculations

FAQs:

What Are The Basic Properties of Exponent?

Law of Product: am × an = a^m+n
Law of Quotient: am/an = a^m-n
Law of Zero Exponent: a^0 = 1
Law of Negative Exponent: a^-m = 1/a^m
Law of Power of a Power: (a^m)^n = a^mn
Law of Power of a Product: (ab)^m = a^mb^m
Law of Power of a Quotient: (a/b)^m = a^m/b^m

What Is The Inverse of an Exponential Function?

The logarithmic functions are the inverse of the exponential functions.

What Are The Rules of an Exponential Function?

There are three main rules to get the exponential functions which are as follows.

Add the exponents to multiply the two exponent functions with the same base.
Subtract the exponents to divide two exponential functions with the same base.
Multiply the exponents in order to raise the power to the new power.

References:

From the source Wikipedia: Exponential function, Relation to more general exponential functions, Derivatives and differential equations, write expression in exponential form calculator.