Question

How do you find the value of \[{{\log }_{2}}100\]?

Answer 1

Hint: To solve the given question, we will need some properties of logarithm and the value of some special logarithm. They are as follows, the property of logarithm we will be using states that, \[{{\log }_{a}}b=\dfrac{{{\log }_{c}}b}{{{\log }_{c}}a}\]. Here \[a,b,\And c\in \] positive real numbers and \[a\And c\] can not take the value one. Also, the property of logarithm which states that, \[\log {{a}^{m}}=m\log a\], here \[a\And m\in \] Real number. The value of the logarithm we will use is of \[\log 2=\text{0}\text{.301}\]. We will use these to find the given logarithm value.

Complete step by step answer:
We are given \[{{\log }_{2}}100\], and we have to find its value. To evaluate this logarithm, we will use the property of logarithm \[{{\log }_{a}}b=\dfrac{{{\log }_{c}}b}{{{\log }_{c}}a}\]. We take the new base c to be 10. So, the given logarithm can be written as,
\[\Rightarrow {{\log }_{2}}100=\dfrac{\log 100}{\log 2}\]
We know that 100 is the square of 10, in the above expression we can replace 100 with \[{{10}^{2}}\]. By doing this, we get
\[\Rightarrow \dfrac{\log {{10}^{2}}}{\log 2}\]
We know the property of logarithm which states that, \[\log {{a}^{m}}=m\log a\]. Using this property in the above expression, we get
\[\Rightarrow \dfrac{\log {{10}^{2}}}{\log 2}=\dfrac{2\times \log 10}{\log 2}\]
As the base and argument for \[\log 10\] are the same its value equals 1. Substituting this in the above expression, we get
\[\Rightarrow \dfrac{2\times \log 10}{\log 2}=\dfrac{2\times 1}{\log 2}=\dfrac{2}{\log 2}\]
We know the value of \[\log 2=\text{0}\text{.301}\]. Using this for the above expression, we get
\[\Rightarrow \dfrac{2}{\log 2}=\dfrac{2}{0.301}\approx 6.64\]
Hence the value of \[{{\log }_{2}}100\] is approximately \[6.64\].

Note:
The value of some special logarithm should be remembered, for example, the value of \[\log 2,\log 3,\ln 10\] equals \[0.301,0.477\And 2.302\] respectively. The properties of logarithms should also be remembered, as they will be useful while solving problems on logarithms.