Question

How do you calculate ${{\log }_{4}}\left( \dfrac{1}{16} \right)$?

Answer 1

Hint: We will look at the definition of a logarithmic function. We will see the relation between the logarithmic function and the exponential function. We will convert the given logarithmic function into its exponential equivalent. After that we will solve the obtained exponential equation to get the solution of the given equation.

Complete answer:
We define the logarithmic function as the inverse of the exponential function. The definition gives us a relation between the logarithmic function and the exponential function. If we have an exponential function as ${{a}^{x}}=b$ then its logarithmic equivalent is given as ${{\log }_{a}}b=x$.
The given logarithmic function is ${{\log }_{4}}\left( \dfrac{1}{16} \right)$. Let ${{\log }_{4}}\left( \dfrac{1}{16} \right)=x$. Now, using the above definition and relation, we will convert this logarithmic function into its exponential equivalent. So, we get the following expression,
${{4}^{x}}=\dfrac{1}{16}$
Now, we know that the square of 4 is 16, that is, ${{4}^{2}}=16$. We also have the rule for exponential functions having negative exponents as ${{a}^{-1}}=\dfrac{1}{a}$. So, combining these two facts, we have that ${{4}^{-2}}=\dfrac{1}{16}$. Comparing this equation with the above equation, we get that $x=-2$. Thus, we have obtained the value of the given expression as ${{\log }_{4}}\left( \dfrac{1}{16} \right)=-2$.

Note: The logarithmic function and the exponential function are very important functions. Their relation is very useful in simplifying and solving equations. There are multiple laws or rules for simplifying expressions that contain these two types of functions. We should be familiar with these as they are very helpful in solving such types of questions.