Question

How do you write the equation $ {{\log }_{4}}\left( \dfrac{1}{4} \right)=-1$ in exponential form?

Answer 1

Hint: In this question, we are given an equation in terms of logarithm and we need to change it into exponential form. For this, we will use the following conversion.
If we have the logarithm with base b and the argument (angle) as x equal to n then we can say that the argument x is equal to the exponential form having base b and power of base b as n i.e. $ {{\log }_{b}}x=n\Rightarrow x={{b}^{n}}$ .

Complete step by step answer:
Here we are given the equation in logarithmic form as $ {{\log }_{4}}\left( \dfrac{1}{4} \right)=-1$ .
We need to convert this equation into the exponential form i.e. we want an equation of the form $ {{x}^{y}}=z$ where $ {{x}^{y}}$ is the exponential term which gives the solution as z. The conversion from logarithm to exponential is done as follows:
If we have the logarithm with base b and the argument (angle) as x equal to n, then we can say that x is equal to the exponential form having base b and the power of b is b as n. In mathematical form, we can say that if $ {{\log }_{b}}x=n$ then in exponential form it can be written as $ x={{b}^{n}}$ .
Let us compare $ {{\log }_{b}}x=n$ with the given equation to apply the conversion.
Comparing $ {{\log }_{b}}x=n$ with $ {{\log }_{4}}\left( \dfrac{1}{4} \right)=-1$ we can see that, base b = 4, argument (angle) $ x=\dfrac{1}{4}$ and value of n is equal to -1.
As $ {{\log }_{b}}x=n$ is changed to $ x={{b}^{n}}$ . So $ {{\log }_{4}}\left( \dfrac{1}{4} \right)=-1\Rightarrow \dfrac{1}{4}={{4}^{-1}}$ .
As we can see $ \dfrac{1}{4}={{4}^{-1}}$ is an exponential form of the equation where, if base 4 is raised to the power of -1 then we get the answer as $ \dfrac{1}{4}$ which is true. Therefore, $ \dfrac{1}{4}={{4}^{-1}}$ is the required equation in the exponential form.

Note:
At last, students should always check if the equation actually holds or not. Students can get confused between base, angles, and values when converting. In $ \log {{b}^{x}}$ , x can also be called angle or power.