Question

How do you evaluate ${\log _{16}}4?$

Answer 1

Hint:In order to solve this question, firstly we have to understand what logarithm of a number means. If the written logarithm of $a$ in base $b$ equals $c$, then it means $a$ is equal to $c$ raised to the power of $b$. Understanding this mathematically as follows
${\log _b}a = c \\
\Rightarrow {b^c} = a $
Use this information to evaluate the given logarithm.

Complete step by step answer:
In order to evaluate ${\log _{16}}4$, let us understand first what logarithm means in a general way and the function of a logarithmic function.Consider a logarithm of $a$ with base $b$ equal to $c$, it means that when $c$ is raised to the power of $b$ then it will equal to $a$, let’s understand this in mathematical way or in equations
${\log _b}a = c$
It is the mathematical form of the first part of the consideration. Now coming to second part
${b^c} = a\; - - - - - - ({\text{i}})$
It is the mathematical form of the second part.
Hope that you got to understand the logarithmic function.

Now let’s come to the question, that is ${\log _{16}}4$ or logarithm of $4$ with base $16$.Let us consider ${\log _{16}}4$ equals to $x$, then it will be written in mathematical equation as follows,
${\log _{16}}4 = x$
Now writing this equation as equation (i) is written, we get
${16^x} = 4$
Solving this further, by converting $16$ into multiples of $2$,
${\left( {2 \times 2 \times 2 \times 2} \right)^x} = 2 \times 2 \\
\Rightarrow {2^{4x}} = {2^2} \\ $
Now comparing the power of both sides, because their bases are same i.e. $2$ we get,
$4x = 2 \\
\Rightarrow x = \dfrac{2}{4} \\
\Rightarrow x = \dfrac{1}{2} \\ $
We got the value of $x$that means,
$ \therefore {\log _{16}}4 = \dfrac{1}{2}$

Note:Logarithm has some strict rules with it, that is the base of logarithm should be greater than $0$ and should not be equal to $1$, also logarithmic functions never take negative arguments. These limits are sometimes used to solve logarithmic problems.