Introduction to Exponential Functions

As the name implies, the exponential function contains an exponential. However, an exponential function has a constant as the base and a variable as the exponential, but not the other way around (if the function has a variable as the base and a constant as the exponential, it is a power function, but it is not). Exponential). The exponential function has one of the following forms.

Exponential Functions

In mathematics, an exponential function is a function of the form f (x) = ax. Where “x” is a variable and “a” is a constant called the base of the function, which must be greater than 0.

Exponential functions are one of the most important functions in mathematics (although it must be acknowledged that linear functions are even more important). To create an exponential function, make the independent variable an exponential.

The exponential function is somewhat similar to the functions we saw earlier in that it contains exponentials, with the major difference in that the variables are exponential rather than base. And the exponents generated by these functions have a “doubling period”, and if you wait long enough, they will grow insanely fast.

Exponential Function Formula

The exponential function is defined by the expression f (x) = ax. Here, the input variable x is displayed as an exponent. The exponential curve is determined by the exponential function and x. The exponential function is an important mathematical function of the following form: 

 f (x) = ax 

 where a> 0 and a are not equal to 1. 

x is any real number. 

 If the variable is negative, the function of 1 & lt; x & lt; 1.1. 

 Where 

 “x” is a variable. 

 “a” is the base constant of the function. 

 The exponential curve increases or decreases depending on the exponential function. An amount that increases or decreases at a constant rate at regular intervals should indicate either exponential growth or exponential decay.

Compound interest

 One of the most common uses of the exponential function is compound interest and compound interest calculations. This discussion focuses on the use of compound interest.

A=P ( 1+ r n ) nt

Where A is the account balance, P is the principal or starting value, r is the annual interest rate as a decimal, n is the annual compounding number, and t is the annual time.

Loudness of sound

Loudness is observed to be a logarithmic function. Therefore, you can use the various properties of the logarithmic function to find the intensity. Since the exponential function is the inverse of the logarithmic function, the various properties of the logarithmic function are due to the properties of the exponential function.

The formula for loudness using sound intensity is:

dB-d𝐵0 = 10 log (I / I0)

Where dB is the loud decibel value, d𝐵0 decibel. The value of soft sound. (I / I0) is the tone intensity ratio.

Population increase

With exponential growth, the growth rate of the population per capita (per capita) remains the same regardless of the size of the population, and the population grows faster and faster as the size increases. Exponential growth is growth that can be modelled with an exponential function. Exponential growth as a characteristic of the doubling period. For example, bacterial colonies in Petri dishes can double every 6 hours. Here, 6 hours is doubling time. Exponential growth generally begins slowly, but once it grows, it grows very rapidly. The term “exponential growth” is often used informally in conversations, news, etc. to describe “really, really fast” growth that may not have a doubling period. Notice this difference. Mathematics has an accurate definition. In general terms, meaning is more fluid. So, when you hear someone claim that the world’s population is doubling every 30 years, you know they are claiming exponential growth.

Exponential growth is “larger” and “faster” than polynomial growth. This means that a particular exponent will eventually be larger than a polynomial, regardless of the degree of the particular polynomial. The exponential function starts from a very small one, but in the end, it definitely overtakes the growth of the polynomial, zooming in on the past and always doubling.

CONCLUSION

The derivative (rate of change) of the exponential function is the exponential function itself. More generally, a function with a rate of change proportional to (rather than equal to) the function itself can be represented by an exponential function. This property leads to exponential growth or exponential decay.