Question

How do you evaluate \[{\log _2}32\]?

Answer 1

Hint: In the given question, we have been given an expression. This expression contains a function, which is the logarithm function. We have to solve the logarithm, when we have been given the base and the argument of the logarithm. We do that by expressing the argument in terms of the given base and solving it by using the appropriate formula.

Complete step by step answer:
We are going to use the formula of logarithm, which is:
\[{\log _b}a = n \Rightarrow {b^n} = a\]
The given expression is \[p = {\log _2}32\]. We have to solve for the value of \[p\].
The basic formula of logarithm says,
If \[{\log _b}a = n\]
then, \[{b^n} = a\]
Hence, putting \[b = 2\], \[a = 32\], we get,
\[{2^p} = 32\]
We know, \[32 = {2^5}\].
Hence, \[{2^p} = {2^5}\]
Since the bases are equal, we can equate the powers,
\[p = 5\]

Hence, \[{\log _2}32 = 5\]

Note: The \[\log \] function has other basic properties too:
\[{\log _x}{x^n} = n\]
\[{\log _a}b = \dfrac{1}{{{{\log }_b}a}}\]
In the given question, we had been given a logarithmic expression. We had to solve the logarithm, when we were given the base and the argument of the logarithm. We solve these questions by applying the appropriate formula. Students make mistakes if they do not know the correct formula and its use cases. So, it is really important that we know all the formulae and their related properties.