Question

How do you calculate \[\log 32\]?

Answer 1

Hint: The logarithm is defined as the power to which a number must be raised to obtain a particular number. For example \[{\log _{10}}100 = 2\] defines that \[10\] must be raised to power \[2\] to obtain the number \[100\]. The other example is \[{\log _2}8 = 3\]show that \[2\] must be raised to power \[3\] to obtain \[8\].
Here the objective is to calculate\[\log 32\]. To evaluate \[\log 32\] write \[32\] in the exponential form with the base of\[2\]. When the base of the logarithm is not given the base is considered naturally\[2\]. Then, apply the property of the logarithm to find the complete value of the\[\log 32\].

Complete step by step solution:
The given expression is c \[\log 32\].
Write the value of \[32\] in the exponential form as a multiple of 2.
We know that \[2\] multiplied \[5\] times by itself is\[32\]. This in the exponential form is written as \[32 = {2^5}\]
Then, the given term \[\log 32\] becomes\[\log {2^5}\].
Consider the property of logarithm as,
\[{\log _a}{b^x} = x{\log _a}b\]
Then, from the above property \[{\log _2}{2^5}\]is written as\[\log {32^5} = 5\log 2\].
Consider the value of \[\log 2 = 0.3010\].
Then, rewrite the logarithm \[\log {32^5} = 5\log 2\] as,
\[ \Rightarrow 5\log 2 = 5\left( {0.3010} \right)\]
\[\therefore 5\log 2 = 1.505\]

Thus, the value of \[\log 32 = 1.505\]

Note:
The exponent is the number of times a number can be multiplied by itself. For example consider a variable $a$ as $a \times a = {a^2}$, then ${a^n}$ represents $a$ multiplied by itself n number of times. The exponent form ${a^n}$ is pronounced as a raise to the power n. Where,
${a^0} = 1$ And ${a^1} = a$.