Question

How do you solve \[{{\log }_{x}}\left( \dfrac{1}{256} \right)=-4\]?

Answer 1

Hint: From the given question we are asked to find the solution for the logarithmic equation. For solving this question we will use the definition of logarithm and we will remove the log present in the left hand side of the equation and we will use basic mathematical operations like division and multiplication and solve the given question.

Complete step by step solution:
We are given that \[{{\log }_{x}}\left( \dfrac{1}{256} \right)=-4\].
From the definition of a logarithm we can remove the log on the left hand side of the equation by bringing the base of the logarithm to the other side of the equation, while the term on the other side is raised to the power of the base.
For example,
\[\Rightarrow {{\log }_{x}}y=t\]
This can be rewritten as follows using the definition of logarithm.
\[\Rightarrow y={{x}^{t}}\]
So, we now apply this same to the given equation on the question.
So, we get the equation reduced as follows.
\[\Rightarrow {{\log }_{x}}\left( \dfrac{1}{256} \right)=-4\]
We will bring the base which is x to the other side of the equation. So, the negative of four will become the power to x.
So, the equation will be reduced as follows.
\[\Rightarrow {{x}^{-4}}=\left( \dfrac{1}{256} \right)\]
We can rewrite the term in the right hand side of the equation as follows.
\[\Rightarrow {{x}^{-4}}=\left( \dfrac{1}{{{4}^{4}}} \right)\]
\[\Rightarrow {{x}^{-4}}={{4}^{-4}}\]
We can now equate the base terms as powers are equal. So, we get the following.
\[\Rightarrow x=4\]

Note: Students must be very careful in doing the calculations. Students should have good knowledge in the concept of logarithm and its applications. We should not do mistakes like for example, if we write \[\Rightarrow {{x}^{-4}}=\left( \dfrac{1}{{{2}^{8}}} \right)\] \[\Rightarrow {{x}^{-4}}={{2}^{-8}}\] and directly equate the base then the solution is wrong.