- The following are exponential forms of some numbers? 10,000 = 10 4 (read as 10 raised to 4) 243 = 3 5, 128 = 2 7. Here, 10, 3 and 2 are the bases, whereas 4, 5 and 7 are their respective exponents. We also say, 10,000 is the 4th power of 10, 243 is the 5th power of 3, etc. 3. Numbers in exponential form obey certain laws, which are:
Remark The exponential series involving variable x can be expressed as Example 3 Find the coefficient of x2 in the expansion of e2x+3 as a series in powers of x. Solution In the exponential series = replacing x by (2x + 3), we get = Here, the general term is = . This can be expanded by the Binomial Theorem as Here, the coefficient of x2 is .
It is advised that the reader plots this graph for particular values of b like 2, 3 and 4. Following are some of the salient features of the exponential functions: (1) Domain of the exponential function is R, the set of all real numbers. (2) Range of the exponential function is the set of all positive real numbers.
In most systems, the initial growth is slow (lag phase), and it increases rapidly thereafter – at an exponential rate (log or exponential phase). Here, both the progeny cells following mitotic cell division retain the ability to divide and continue to do so. However, with limited nutrient supply, the growth slows down leading to a stationary ...
In Chapter 5, we have learnt how to find derivative of composite functions, inverse trigonometric functions, implicit functions, exponential functions and logarithmic functions.