Question

How do you simplify ${{5}^{{{\log }_{5}}x}}$ ?

Answer 1

Hint: We know the concept of logarithm if ${{\log }_{a}}b=c$ then we can write $b={{a}^{c}}$. In the first equation we write the value of c is equal to ${{\log }_{a}}b$ so we can replace c in the equation we get $b={{a}^{{{\log }_{a}}b}}$ we can use this property to solve the above question.

Complete step by step answer:
The given equation we have to simplify is ${{5}^{{{\log }_{5}}x}}$
We know the property of logarithm if $b={{a}^{c}}$ then we can replace c with ${{\log }_{a}}b$ and can write $b={{a}^{{{\log }_{a}}b}}$ where a and b should not be negative number because the domain of logarithm function is always positive number
Applying above statement to the equation given in the question we can write ${{5}^{{{\log }_{5}}x}}$ is equal to x and x should not be a negative number
If x is negative number then the function ${{\log }_{5}}x$ will be invalid so the function ${{5}^{{{\log }_{5}}x}}$ will also be invalid.

Note:
We can understand the above question from another point let’s take a function $f\left( x \right)={{5}^{x}}$, we can see that the range of the function is always positive. Let’s find out the inverse function of ${{5}^{x}}$ so the inverse of ${{5}^{x}}$ is ${{\log }_{5}}x$ . Range of ${{5}^{x}}$ is equal to domain of ${{\log }_{5}}x$ so the domain of ${{\log }_{5}}x$ is always positive let’s take ${{\log }_{5}}x=g\left( x \right)$ . We know that if f and g are inverse functions of each other then we can write $f\left( g\left( x \right) \right)=x$ . If the $f\left( x \right)={{5}^{x}}$ and the value of $g\left( x \right)$ is ${{\log }_{5}}x$ then $f\left( g\left( x \right) \right)$ is ${{5}^{{{\log }_{5}}x}}$ . We know that $f\left( g\left( x \right) \right)=x$ so the value of ${{5}^{{{\log }_{5}}x}}$ is x .