Question & Answer

What is the value of ${\log _2}64$?
A. 2
B. 3
C. 4
D. 6

ANSWER Verified Verified
Hint: Convert 64 into the powers of 2 so that we can apply the property of log that is ${\log _b}{b^x} = x$ to get the correct answer to this question.

Complete step-by-step answer:

Given that, ${\log _2}64$
Now, we can solve this 64 by taking it as a power of the base of log i.e. 64 can be written as a power of 2
Powers of 2 are like this:
  {2^1} = 2 \\
  {2^2} = 4 \\
  {2^3} = 8 \\
  {2^4} = 16 \\
  {2^5} = 32 \\
  {2^6} = 64 \\
We can raise 2 to the power of 6 in order to 64. Instead of 64 we will write ${\log _2}{2^6}$,
Now, we can write logarithm as ${\log _2}{2^6}$
As we know, that one property of logarithm is ${\log _b}{b^x} = x$ i.e. if the base of the logarithm is equal to the number given in $\log $, we will get the answer as the exponent or power of that number.
By using, this property we get the answer as ${\log _2}{2^6}$$ = 6$, where base$\left( {_b} \right)$ is 2, the number given in the $\log $ i.e. $\left( b \right)$ is 2 and the exponent or power of that number i.e. $\left( {^x} \right)$ is 6 and according to the particular property, now we can say that the answer should be the value of $x$ i.e. 6.
Hence, the answer is 6
So, the correct option is D.

Note: Remember the properties of logarithm and identify which one to use in the questions by realising the format of the question. Remember to calculate the powers of 2 carefully as one can make an error in the calculations. You can also follow another method by factoring 64 into $8 \times 8$ and then by applying the log of a product property on the answer but at the end you will need to write 8 in terms of power of 2 as well.