Question

The value of ${\text{lo}}{{\text{g}}_{\text{2}}}{\text{16}}$ is
A. $\dfrac{1}{4}$
B. 4
C. 8
D. 16

Answer 1

Hint: Logarithms are another way of thinking about exponents. For example, we know that 2 raised to the 4th power equal 16. This is expressed by the exponential equation ${2^4} = 16$. The above question is the same, expressed by the logarithmic equation.

Complete step-by-step answer:

We have,
 ${\text{lo}}{{\text{g}}_{\text{2}}}{\text{16}}$
In this logarithmic equation 2 is the base and 16 is called the argument.

We use the property of logarithm.
${\text{lo}}{{\text{g}}_{\text{a}}}{{\text{a}}^{\text{n}}}{\text{ = n lo}}{{\text{g}}_{\text{a}}}{\text{a}}$ and
${\text{lo}}{{\text{g}}_{\text{a}}}{\text{a = 1}}$

Now, we express ${2^4} = 16$.
${\text{lo}}{{\text{g}}_{\text{2}}}{\text{16}}$ = ${\text{lo}}{{\text{g}}_{\text{2}}}{{\text{2}}^{\text{4}}}$
               = 4 ${\text{lo}}{{\text{g}}_{\text{2}}}{\text{2}}$
               = 4 $ \times $ 1
               = 4

Therefore, the value of ${\text{lo}}{{\text{g}}_{\text{2}}}{\text{16}}$ is 4.
So, option (B) is correct.

Note: When rewriting an exponential equation in log form or a log equation in exponential form, it is helpful to remember that the base of the logarithm is the same as the base of the exponent. ${\text{lo}}{{\text{g}}_{\text{a}}}{\text{a}}$ is defined when the base a is positive. And the natural logarithm is a logarithm whose base is the number ‘e’.