Question

How do you evaluate \[\sqrt {\log 2 \cdot 20 - \log 16} \] ?

Answer 1

Hint: Logarithmic functions are the inverses of exponential functions. In Logarithms, the power is raised to some numbers (usually, base number) to get some other number. It is an inverse function of exponential function. Here in the given expression, we need to evaluate the log values given in the square root, hence apply the logarithmic properties and simplify the terms of the logarithm function to get the value of the square root.
Formula used:
 \[\log a - \log n = \log \left( {\dfrac{a}{n}} \right)\]
 \[a\log n = \log {n^a}\]

Complete step-by-step answer:
Given,
 \[\sqrt {\log 2 \cdot 20 - \log 16} \]
The given, expression is written as:
 \[ \Rightarrow \sqrt {20\log 2 - \log 16} \] ………………….. 1
To evaluate the given expression, we need to apply logarithmic rule as the expression is of the form \[a\log n\] , hence we know that; \[a\log n = \log {n^a}\] .
Hence, applying this to the equation 1 we get:
 \[\sqrt {20\log 2 - \log 16} \]
 \[ \Rightarrow \sqrt {\log \left( {{2^{20}}} \right) - \log 16} \]
We, know that: \[\log \left( {{2^{20}}} \right) = 10,48,576\] , hence we get:
 \[ \Rightarrow \sqrt {\log 10,48,576 - \log 16} \] …………………… 2
Now, the obtained equation 2 is of the form, \[\log a - \log n\] , hence let us apply the rule as: \[\log a - \log n = \log \left( {\dfrac{a}{n}} \right)\] , we get:
 \[ \Rightarrow \sqrt {\log \dfrac{{1048576}}{{16}}} \]
Evaluating we get:
 \[ \Rightarrow \sqrt {\log \left( {65536} \right)} \]
We have the value of \[\log \left( {65536} \right)\] as:
 \[ \Rightarrow \sqrt {4.81658} \]
Hence, the square root of 4.81658 is:
 \[ \Rightarrow 2.19\]
Therefore,
 \[\sqrt {\log 2 \cdot 20 - \log 16} = 2.10\] .
So, the correct answer is “2.10”.

Note: The key point to evaluate the given expression, is that we must know all the log properties. We must know that, logarithms to the base 10 are referred to as common logarithms. When a logarithm is written without a subscript base, we assume the base to be 10. Logarithms to the base ‘e’ are called natural logarithms. The constant e is approximated as 2.7183. Natural logarithms are expressed as ln x which is the same as log e. The logarithmic value of a negative number is imaginary and the logarithm of any positive number to the same base is equal to 1.