An exponential equation is an expression where both sides can be presented in the form of same based and it can be solved with the help of a property. It is generally used to express a graph in many applications like Compound interest, radioactive decay, or growth of population etc. The general form of an exponential equation includes –

\[\large y=a^{x}\]

Where a are the constants and x, y are the variables.

The exponential function is a special type where the input variable works as the exponent. A function f(x) = bx + c or function f(x) = a, both are the exponential functions. It is used everywhere, if we talk about the C programming language then the exponential function is defined as the e raised to the power x. Here, x could be any real number. The syntax for exponential functions in C programming is given as –

The mean of the **Exponential (λ) Distribution** is calculated using integration by parts as –

\[\large E(X) = \int_{0}^{\infty } x\lambda e^{-\lambda x} \; dx\]

\[\large = \lambda \left [ \frac{-x \; e^{-\lambda x}}{\lambda}|_{0}^{\infty } + \frac{1}{\lambda }\int_{0}^{\infty } e^{-\lambda x} dx \right ]\]

\[\large = \lambda \left [ 0 + \frac{1}{\lambda }\frac{-e^{-\lambda x}}{\lambda} |_{o}^{\infty }\right ]\]

\[\large = \lambda \frac{1}{\lambda ^{2} }\]

\[\large = \frac{1}{\lambda }\]

double exp (double x);

Parameters or Arguments – X

In case, the magnitude of the variable is too large then it may throwback an error. Understanding exponential functions are not easy but it is necessary when they are needed to use for the real-life applications.

The exponential distribution in probability is the distribution that explains the time among events in a Poisson process. Further, we will discuss the exponential growth and exponential decay formulas and how can you use them practically.

Exponential growth is the condition where the growth rate of the mathematical function is directly proportional to the current value of the function that results in growth with time being an exponential function. The Exponential growth formula in mathematics is given as –

Formula of Exponential Growth

\[\large P(t)=P_{0}e^{rt}\]

Where:

t = time (number of periods)

P(t) = the amount of some quantity at time t

P_{0} = initial amount at time t = 0

r = the growth rate

Where t is the time (total number of periods), P(t) is the amount of a quantity at given time t, P0 is the initial among at the time t = 0, and r is taken as the growth rate.

Do you know the fact that most of the graphs have the same arcing shape? Any graph could not have a constant rate of change but it may constant ratios that grows by common factors over particular intervals of time.

- All the domains are the real numbers.
- The value will be positive numbers, not the zero.
- For a real number having power zero, the final value would be one.
- If b is the base whose value is greater than one then graph will increase. The greater the value of b, the faster the graph will increase from left to right.
- If b is the base whose value is less than one then the graph will decrease. The lower is the value of b, the graph will increase from right to right.

There are two popular cases in case of Exponential equations. These are the exponential growth and the exponential decay. In the case of Exponential Growth, quantity will increase slowly at first then rapidly. In the case of exponential decay, the quantity will decrease faster at first then it will move slowly. Their formulas can be given as shown below:

\[\large y=ab^{x}\]

Here,

x and y are the variables

a and b are constants.