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How do you find the domain and range of $y={{\log }_{5}}x$?

Last updated date: 13th Jun 2024
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Hint: The logarithm is used to convert a large or very small number into the understandable domain. For the theorem to work the usual conditions of logarithm will have to follow. We also need to remember that for logarithm function there has to be a domain constraint. The range in the usual case is the whole real line.

Complete step-by-step solution:
Let an arbitrary logarithmic function be $A={{\log }_{b}}a$. The conditions for the expression to be logical is $a,b>0;b\ne 1$.
We have $\log {{x}^{a}}=a\log x$. The power value of $a$ goes as a multiplication with $\log x$.
In case of logarithmic numbers having powers, we have to multiply the power in front of the logarithm to get the single logarithmic function.
Let’s assume $y=f\left( x \right)={{\log }_{5}}x$.
We need to find the domain and range of $y={{\log }_{5}}x$.
Using the previous mentioned conditions, we can say that $x>0$.
The domain of the function is $\left( 0,\infty \right)$.
The range of any logarithm function is the whole real line.
Therefore, for the function $y={{\log }_{5}}x$, the range is $\mathbb{R}$.

Note: There are some particular rules that we follow in case of finding the condensed form of logarithm. We first apply the power property first. Then we identify terms that are products of factors and a logarithm, and rewrite each as the logarithm of a power. Then we apply the product property. Rewrite sums of logarithms as the logarithm of a product. We also have the quotient property rules.