Question

How do you solve the equation $ {\log _x}243 = 5 $

Answer 1

Hint: The logarithmic number is converted to the exponential number. The exponential number is defined as the number of times the number is multiplied by itself. The logarithmic number has a base and we have to find the value of base by conversion.

Complete step-by-step answer:
The given number is in the form of a logarithmic number and we have to convert it into exponential form. The equation is in the form $ {\log _x}y = b $ to convert it into exponential form it is written as $ y = {x^b} $ , where x is the base of the log function.
Consider the given question $ {\log _x}243 = 5 $ , when we compare to general form y is 243 and b is 5. Therefore, it is written as $ 243 = {x^5} $
The number 243 is factorised as

3	243
3	81
3	27
3	9
3	3
	1

Therefore, the number 243 is written in the form of exponential form as $ 3 \times 3 \times 3 \times 3 \times 3 = {3^5} $
The above equation is written as
 $ \Rightarrow {3^5} = {x^5} $
The power of the number is the same then the value of the base number is also the same.
Therefore, the value of x is $ x = 3 $
Hence, we solved the equation $ {\log _x}243 = 5 $ and determined the value of x that is $ x = 3 $
So, the correct answer is “ $ x = 3 $ ”.

Note: To solve the logarithmic equation we need to convert the equation to the exponential form and by using the concept of factorisation we can determine the value of the base of the log. The exponential form of a number is defined as the number of times the number is multiplied by itself. The general form of logarithmic equation is $ {\log _x}y = b $ and it is converted to exponential form as $ y = {x^b} $ . Hence we obtain the result or solution for the equation.