Question

Express the following in exponential form: ${{\log }_{9}}6561=4$.

Answer 1

Hint:We will use the concept ${{\log }_{b}}x=y$ can be written as ${{b}^{y}}=x$ such that $x>0,b>0$ and also $b\ne 1$. In this case we have $b=9$ and $x=6561$ also $y=4$.So, use this concept and get the answer.

Complete step-by-step answer:
It is given in the question that we have to express the given logarithm, that is, ${{\log }_{9}}6561=4$ into exponential form. For this, we will use the basic formula and concept of logarithms that ${{\log }_{b}}x=y$ can be written as ${{b}^{y}}=x$ such that $x>0,b>0$ and also $b\ne 1$.
Now, in the question we are given ${{\log }_{9}}6561=4$. So, here we have $b=9$ and $x=6561$ also $y=4$.
On using the above concept and substituting the value of b,x and y into the equation ${{b}^{y}}=x$, we get
${{b}^{y}}=x$
${{9}^{4}}=6561$
Thus 6561 can be represented as ${{9}^{4}}$ and we can rewrite the given logarithm as $6561={{9}^{4}}$.

Note: Students should know the concept of logarithmic equation ${{\log }_{b}}x=y$ which can be written as ${{b}^{y}}=x$ such that $x>0,b>0$ and also $b\ne 1$. Sometimes, in some tricky questions, the latter conditions play an important role in deciding the answer to the question, mere conversion from ${{\log }_{b}}x=y$ to ${{b}^{y}}=x$ is not sufficient to obtain the answer. Therefore, while remembering the conversion, we must also remember the side conditions mentioned along the main conversion.Students can make mistakes by taking ${{y}^{b}}=x$ instead of ${{b}^{y}}=x$ can lead to wrong answers So be careful while writing the logarithmic equation.We can also check L.H.S =R.H.S after expressing it into exponential form.