Question

What is ${{\log }_{3}}243$?

Answer 1

Hint: To solve this question we can consider the following property of logarithm:
The logarithm of a number such that it has the same base as the number is 1 which can be given as:
${{\log }_{a}}a=1$
If the number is raised to some power and the base and number re same then the value is equal to the power, which is given as:
${{\log }_{a}}{{a}^{x}}=x$
We can write the expression involving log as an algebraic expression as follows:
 $\begin{align}
  & x={{\log }_{a}}b \\
 & \Rightarrow {{a}^{x}}=b \\
\end{align}$

Complete step by step answer:
Given,
${{\log }_{3}}243$
To find the value of ${{\log }_{3}}243$ .
Consider the value of function equal to $x$ therefore we can write the function as:
$x={{\log }_{3}}243$ …(1)
By using the property of logarithm that if we have,
$\begin{align}
  & x={{\log }_{a}}b \\
 & \Rightarrow {{a}^{x}}=b \\
\end{align}$
Therefore, using this in equation (1) the value of a is 3 and that of b is 243 we get:
${{3}^{x}}=243$ …(2)
Now we can write 243 such that the base and number are same or express it as a power of 3 thus from equation (1) we get:
$243={{3}^{5}}$ …(3)
Now using the equation (2) and (3) we can write the (4) equation as follows:
${{3}^{x}}={{3}^{5}}$ …(4)
Now we can compare powers at L.H.S and R.H.S from equation (4) we get:
$x=5$

Thus, the value of ${{\log }_{3}}243$ is $5$.

Note: We can solve this question by another method where we can use the property:
Consider the function as follows:
$x={{\log }_{3}}243$ ..(1)
If the number is raised to some power and the base and number re same then the value is equal to the power, which is given as:
${{\log }_{a}}{{a}^{x}}=x$
Thus, we can write ${{\log }_{3}}243$ by expressing the number 243 in terms of powers of 3 we have:
$243={{3}^{5}}$ ..(2)
Thus, we can write equation (1) as:
$x={{\log }_{3}}{{3}^{5}}$
Same base and power are cancelled hence we have,
$x=5$