# Exponential Growth Formula

## Exponential Growth Formula

Algebra is a broad area of mathematics. In a nutshell, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a common thread that runs through almost all Mathematics.

In Mathematics, Exponential Growth Formula and formulas aid in the calculation of large numbers and are applied in various real-world situations. For example, we can use the Exponential Growth Formula to calculate the population growth of a city, the rate of change of bacteria in a culture, the half-life, the radioactive decay of radioactive isotopes, and so on. An Exponential Growth Formula, as the name implies, is a function that involves exponents. There are two types of Exponential Growth Formulas: Exponential Growth Formula and exponential decay. We will discuss the definition of an exponential function, its graph, types, and exponential formulas, as well as some solved examples, in this article.

An Exponential Growth Formula, as the name implies, is a function that involves exponents. A mathematical function is written as f(x) = axe, where “a” is the function’s base, which is a constant greater than zero, and “x” is the function’s exponent, which is a variable. When x > 1, the function f(x) grows as x gets more extensive. Typically, the base of an Exponential Growth Formula is a transcendental number denoted by e. “e” has a value of approximately 2.71828. An Exponential Growth Formula curve is affected by the value of x. An exponential function’s domain is a set of all real numbers R, whereas its range is a set of all positive real numbers.

## Meaning of Exponential Growth Formula

As the name implies, in exponential growth, a quantity grows slowly at first and then rapidly. The graph of an exponentially growing function is increasing. The Exponential Growth Formula can be used to depict economic growth, population expansion, compound interest, bacterial growth in culture, population increases, and so on.

As the name implies, in exponential decay, a quantity decreases rapidly at first and then gradually fades. The graph of an exponentially decaying function is decreasing. The concept of exponential decay can be used to calculate half-life, mean lifetime, population decay, radioactive decay, and other parameters.

An Exponential Growth Formula is a mathematical function of form f (x) = ax, where ‘x’ is variable and ‘a’ is a constant that must be greater than 0. The most commonly used Exponential Growth Formula basis is the transcendental wide variety e, which is approximately equal to 2.71828.

### Formula of Exponential Growth

Exponents, as the name implies, are used in Exponential Growth Formula. A number’s exponent (base) indicates how many times the number (base) has been multiplied. An exponential equation is one in which the power is a variable in and of itself.

In an Exponential Growth Formula, a variable is an exponent (or a part of the exponent). As an example,

3 ^ x = 243

5 ^ (x – 3) = 125 \s6 ^ (y – 7) = 216

The examples above are of Exponential Growth Formula. Take note of how the variables x and y are either completely or partially from the exponent in the equation. The Exponential Growth Formula is frequently used to solve problems involving compound interest, exponential growth, decay, and so on.

### Solved Examples Using Exponential Growth Formula

1. Solve 5x = 4.

Because the bases in the given equation cannot be made equal, logarithms must be used to solve for x.

⇒ log 5x = log 4

As per the property log am = m log a, we have:

⇒ x log 5 = log 4

Divide both LHS and RHS by log 5.

⇒ x = log 4/log 5.

2. The first prize in a radio station contest is a \$100 gift card. Every day, a name is called. If the person does not contact the company within 15 minutes, the award will be increased by 2.5 per cent the following day. If there are no winners after t days, write an equation to express the gift card’s monetary value.

The equation for exponential growth is y = a(1 + r) ^ t.

We have, a = 100, r = 2.5% or 0.025

In the equation y = 100(1.025) ^ t, y is the amount of the gift card and t is the number of days since the contest began.

3. In 2010, a gym sold 550 memberships. Since then, subscriptions have increased at a rate of 3% per year. Create an equation to represent the number of memberships sold over t years.

The equation for exponential growth is y = a (1 + r) ^ t.

We have, a = 550, r = 3% or 0.03

In the equation y = 550(1.03) ^ t, y is the number of subscriptions sold and t is the number of years.