Chapter 6.3, Problem 4E

To determine

The value of the power of 4 that gives the value 16 and value of ${\log }_{4}16$ then compare the answers.

Answer to Problem 4E

The requires answer is 2 for both the expression.

Explanation of Solution

Given information:

The given expressions are $4^2$ and ${\log }_{2}16$ .

Formula Used:

Logarithmic formula:

  ${\log }_a{a}^b=b$ since ${\log }_{a}a=1$

Consider the first expression $4^2$ .

  $\begin{array}{c}{4}^x=16\\ 4^x=4^2\\ x=2\end{array}$

Thus, 4 to the power 2 gives 16.

Similarly consider the second function ${\log }_{2}16$ .

Now, find the value of ${\log }_{4}16$ . Use the formula ${\log }_a{a}^b=b$ .

  $\begin{array}{c}{\log }_{4}16={\log }_4{4}^2\\ =2{\log }_{4}4\left[{\log }_{4}4=1\right]\\ =2\end{array}$

From the above two solution it is observe that both power form and the logarithm are inverse of each other.

Hence, the requires answer is 2 for both the expression.