Question

What is the exponential form of the logarithmic equation?

Answer 1

First of all we will understand the terms ‘exponentiation’ and ‘logarithm’. Now, we will see their properties and the relation between them that is required for inter – conversion of functions. We will consider some examples in which we will convert exponential functions into logarithmic functions.

Complete step-by-step solution:
Here we have been asked how we convert a logarithmic form of an equation into the exponential form. First we need to understand the terms exponential functions and logarithmic functions.
(1) Exponentiation: - In mathematics exponentiation is an operation that involves two numbers ‘a’ and ‘b’ such that it is denoted as ${{a}^{b}}$ and it denotes that ‘a’ is multiplied with itself b times. Here, ‘a’ is called the base while ‘b’ is called the exponent. For example: - ${{2}^{3}}$ means 2 must be multiplied 3 times so the result will be $2\times 2\times 2=8$.
(2) Logarithm: - In mathematics logarithm is the inverse function of exponentiation. That means that the logarithm of a given number (argument) ‘n’ is the exponent to which another fixed number (base) ‘a’ must be raised to produce that number ‘n’. For example: - if we write ${{\log }_{a}}n=b$ then it means that ‘a’ must be raised to the power ‘b’ to get the value n. Here, ‘a’ and ‘n’ cannot be negative or 0 also ‘a’ cannot be 1.
Now, the relation between a log function and an exponential function is given as if ${{\log }_{a}}n=b$ then ${{a}^{b}}=n$. Let us take an example of a log function given as ${{\log }_{10}}y=x$ and we have to convert it into the exponential form. So using the relation between the two functions we can write: -
$\Rightarrow y={{10}^{x}}$
Hence, the above relation is the exponential form.

Note: Note that in mathematics we come across two types of log namely: - common logarithm and natural logarithm. The base values of both the log functions are fixed. For common log it is 10 and for natural log it is e (also called Euler’s number whose value is nearly 2.71). There are certain properties of logarithms in contrast to the laws of exponents which must be remembered. For example: - ${{\log }_{m}}{{n}^{a}}=a{{\log }_{m}}n$, ${{\log }_{a}}\left( m\times n \right)={{\log }_{a}}m+{{\log }_{a}}n$, ${{\log }_{a}}\left( m\div n \right)={{\log }_{a}}m-{{\log }_{a}}n$ etc.