![]() ![]() Does it make sense to have negative value of time (when we put n < 0)? Indeed, No! Therefore, we must define the ‘allowed’ values of input or the independent variable. Mathematically speaking, it is evident that we can put in any real value of n and should get some value of N as output. Note the significance of the concept of domain here. Figure 1 Graph of exponential funciton representing bacterial growth (at T = 0.4 units) Using calculator or any plotting software (such as desmos), we can get the following plot for growth of bacterial culture. We observe that if we replace x = nT and N = y in the above equation, we get the form of an exponential function (i.e., ) that we are familiar with. The ‘n’ represents the number of time periods T that have passed.Ĭongratulations! We just developed our own exponential function. an exponential function as:Ĭonvince yourself that the bacterial population N is indeed represented by this equation. We can write this process in form of a mathematical formula i.e. After time 2T, the number will increase to and so on. ![]() ![]() Therefore, after the first T time, there will be bacterial cells (on the petri dish). Each bacterial cell divides into two within a time period T. Practically ‘infinite’ resources (food and space) have been provided to support the natural growth of bacteria. Say, we have a bacterial culture on a petri dish, with initially bacterial cells. The practical application of the exponential function is in modelling of a population growth. The corresponding values (output of the function) for all the values in domain (input of the function), form the Range of the exponential function.Īlso, note that means that “ a is an element of the set of real numbers”.All the allowed values of x for the function form the Domain of the exponential function.Change the value of x and you will get corresponding value of y. Independent variable means that we can pick any value of x from the domain (explained next) and pass to the function, that outputs the result in form of y. x is the independent variable, whereas y is the dependent variable.‘x’ is called the exponent or power of the exponential function.‘a’ is called the base of the exponential function and,.It is defined as any function that possess the following form: On the contrary, the concept of ‘ Log’ or ‘ Logarithm’ follows naturally into the minds when we understand the concept of ‘ Exponents’ and ‘ Exponential functions’. Students are often reluctant to explore the beautiful maths behind the logarithm and logarithmic functions. When the graph approaches the y-axis so very closely but will never cross it, we call the line the y-axis, a vertical asymptote.In maths textbooks, whenever we read the word ‘ Log’, it feels like it is some alien idea that is not easy to understand. We write the range in interval notation as What is the range for each function? From the graphs we can see that the range is the set of all real numbers. We write the domain in interval notation as What is the domain of the function? The graph never hits the y-axis. Now we will see that many characteristics of the logarithm function are simply ’mirror images’ of the characteristics of the corresponding exponential function. ![]() Notice too, the graph of each function also contains the point This makes sense as means which is true for any a. The graph of each function, also contains the point This makes sense as means which is true for any a. We notice that for each function the graph contains the point This make sense because means which is true for any a. The graphs of and are the shape we expect from a logarithmic function where ![]()
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