Math Calculators ▶ Exponential Function Calculator
Adblocker Detected
We always struggled to serve you with the best online calculations, thus, there's a humble request to either disable the AD blocker or go with premium plans to use the AD-Free version for calculators.
Disable your Adblocker and refresh your web page 😊
Table of Content
Our exponential function calculator instantly calculates the function that passes from the two points on an x-y plane. Get to know whether the function is exponentially increasing or decaying with respect to time by using this tool.
A function in the form of f(t) = A0e^kt is known to be the exponent function. The exponential function passes through two given points in the x-y plane. You need to provide the values of two points in the x-y plane to calculate the values of the exponent function. The exponential calculator computes the exponent function by inserting the values of (t1,y1) and (t2,y2).
Consider two functions (y1, y2) and their respective values (4, 5) at the relative time (2,5). We want to evaluate their behavior at the time t = 5.
Time 1 (t1): 2
y1 = Function at Time1: 4
Time 2 (t2): 5
y2 = Function at Time2: 5
The time to evaluate = 5
The generic form of the exponential function is:
f(t) = A0e^kt
We need to solve the following equation:
$$ {y_1}={A_0e^{kt_1}}$$
$$ {y_2}={A_0e^{kt_2}}$$
The Exponential function can be evaluated by the following steps:
In the first step, divide y1 and y2 to cancel A0.
$$ \dfrac{y_1}{y_2} = \dfrac{A_0e^{kt_1}}{A_0e^{kt_2}}$$
$$=> \dfrac{y_1}{y_2} = \dfrac{\require{cancel}\cancel{A_0}e^{kt_1}}{\require{cancel}\cancel{A_0}e^{kt_2}}$$
$$=> \dfrac{y_1}{y_2} = \dfrac{e^{kt1}}{e^{kt_2}}$$
You need to evaluate the second step to find the values of k.
$$ \dfrac{y_1}{y_2} = \dfrac{e^{kt_1}}{e^{kt_2}} $$
$$ \dfrac{y_1}{y_2} = e^{kt_1}.e^{kt_2} $$
$$ \dfrac{y_1}{y_2} = e^{k(t_1 – t_2)} $$
$$ In ({\dfrac{y_1}{y_2}}) = In(e^{k(t_1 – t_2)}) $$
$$ In ({\dfrac{y_1}{y_2}}) = e.k(t_1 – t_2) $$
$$ k = \dfrac{1}{t_1 – t_2} In ({\dfrac{y_1}{y_2}}) $$
The exponential calculator can compute the rate of decay and growth of the functions.
$$ A_0e^{kt_1} $$
Or
$$ A_0 = y_1e^{-kt_1} $$
$$ A_0 = y_1e^{-({\dfrac{1}{t_1 – t_2} In ({\dfrac{y_1}{y_2}})})t_1} $$
$$ A_0 = \require{cancel}\cancel{y_1} × \dfrac{y_2}{\require{cancel}\cancel{y_1}e^{kt_2}} $$
$$A_0 = y_2e^{-kt_2}$$
$$ k = \dfrac{1}{2- 5} In ({\dfrac{4}{5}}) $$
k = 0.0744
Now we have:
$$ A_0 = y_2e^{-kt_2}$$
$$A_0 = 5×e^{-0.0744×5}$$
Write an expression in exponential form calculator to evaluate the values of the exponential function and the graphical representation of it.
The final exponential function is:
$$f(t) = A_0e^{kt}$$
$$ f(t) = 3.4468e^{0.0744t}$$
Now you need to analyze the behavior of the exponential function at “5”.
$$f(5) = 3.4468e^{0.0744×5}$$
$$ f\left(5\right) = 5 $$
You can use the exponential equation calculator to validate the results in a matter of seconds.
Using this exponential function formula calculator requires the following inputs to calculate results:
Input:
Output:
The two types of exponential functions are exponential growth and exponential decay. The negative growth is represented by the exponential decay that can also be calculated by the exponential function calculator.
From the source of the wikipedia.org: Exponential function, Graph
From the source of the mathinsight.org: Parameters of the exponential, The exponent