Question

How do you write ${{\log }_{3}}27=x$ in exponential form?

Answer 1

Hint: We will look at the definition of the logarithmic function and the exponential function. Then we will see the relation between these two functions. We will look at an example demonstrating this relation between the two functions. Then we will use this concept to convert the given equation into its exponential form.

Complete step by step answer:
The exponential function expresses a quantity and the number of times it is to be multiplied to itself. The quantity to be multiplied to itself is called the base and the number of times to be multiplied is called the exponent. For example, ${{a}^{x}}$ is an exponential function where $a$ is the base and $x$ is the exponent.
The logarithmic function is defined as the inverse of the exponential function. So, if we have the exponential function as $y={{a}^{x}}$ then its equivalent logarithmic function to this is given as ${{\log }_{a}}y=x$.
For example, we have ${{2}^{3}}=8$. Using the concept given above, we can rewrite this expression in the logarithmic form as ${{\log }_{2}}8=3$.
Now, the given equation is ${{\log }_{3}}27=x$. We can convert this into its exponential form using the concept given above. The exponential form of the given equation is ${{3}^{x}}=27$.

Note: We know that ${{3}^{3}}=27$. Therefore, we can solve the exponential equation obtained and find the value of $x$ as $x=3$. The conversion between logarithmic function and exponential function is very useful in calculations and simplifications. We should be familiar with the working of both these types of functions and their importance.