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
$$\begin{gathered} {\log }_{b}a=c \\ \Rightarrow b^c=a \end{gathered}$$
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 $2$,
$$\begin{gathered} {\left(2\times 2\times 2\times 2\right)}^x=2\times 2 \\ \Rightarrow 2^{4x}=2^2 \end{gathered}$$
Now comparing the power of both sides, because their bases are same i.e. $2$ we get,
$$\begin{gathered} 4x=2 \\ \Rightarrow x={\displaystyle \frac{2}{4}} \\ \Rightarrow x={\displaystyle \frac{1}{2}} \end{gathered}$$
We got the value of $x$that means,
$\therefore {\log }_{16}4={\displaystyle \frac{1}{2}}$

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.