Question

How do you calculate \[{\log _2}512\]?

Answer 1

Hint: Logarithm is the inverse function of exponentiation. To find the value of the given expression, use the logarithmic properties \[{\log _b}{a^x} = x{\log _b}a\] and \[{\log _a}a = 1\]. Try to factorise \[512\] and express it as an exponent of \[2\] as the base of the logarithm is \[2\], then the second property mentioned above can be applied.

Complete step by step solution:
So to find the solution of \[{\log _2}512\], first factorize \[512\]
\[512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\]
\[ \Rightarrow 512 = {2^9}\]
So we can write \[{\log _2}512 = {\log _2}{2^9}\] …equation (1)
Now using the logarithmic identity,
\[{\log _b}{a^x} = x{\log _b}a\]
We can write \[{\log _2}{2^9} = 9{\log _2}2\] … equation (2)
So comparing equation (1) and equation (2), we get
\[{\log _2}512\] \[ = \] \[9{\log _2}2\]
Using the property \[{\log _a}a = 1\], we get
\[{\log _2}2 = 1\]
So, \[{\log _2}512\]\[ = \] \[9 \times 1\]
\[ \Rightarrow \] \[{\log _2}512\] \[ = 9\]

Note: To solve problems like these on logarithm, always try to factorise the given number and express it in the simplest exponential form. Then try to use the properties of logarithm to simplify further. Also note that the base of the logarithm is not always the same always, and the values change according to the base, so note the base carefully while solving. Some of the properties which need to be memorised are mentioned below:
\[{\log _a}1 = 0\]
\[{\log _a}a = 1\]
\[{\log _a}{a^x} = x{\log _a}a\]
\[{\log _c}(ab) = {\log _c}a + {\log _c}b\]
\[{\log _c}\left( {\dfrac{a}{b}} \right) = {\log _c}a - {\log _c}b\]