Question

If $\log 2 = 0.3010$ , then find the value of $\log 200$ ?

Answer 1

Hint : In the above question, we have to find the value of a logarithm expression when we are given the value of its primitive expression from which the required expression can be derived from. So, in order to solve this question, we must know the concepts of logarithms and its properties so that we can express the required logarithmic expression in the terms of given information.

Complete step-by-step answer :
So, in the given question, we have to find the value of $\log 200$ when we are given that the value of $\log 2$ is $0.3010$ .
So, to express $\log 200$ in the terms of known quantities and given information, we must know what logarithm functions are.
A logarithm function is the inverse of an exponential function (a function in which one term is raised to the power of another term is known as an exponential function). An exponential function is of the form $a = {x^y}$ , so the logarithm function being the inverse of the exponential function is of the form $y = {\log _x}a$ .
To evaluate the value of $\log 200$ , we will make use of the value of $\log 2$ given to us in the question and the laws or properties of the logarithm.
So, we have, $\log 200 = \log \left( {2 \times 100} \right)$
Now, we have to expand this expression using the property of logarithm $\log \left( {ab} \right) = \log \left( a \right) + \log \left( b \right)$ .
So, we get,
$\Rightarrow \log 200 = \log \left( 2 \right) + \log \left( {100} \right)$
Now, we know that the value of $\log 100$ is $2$ as ${10^2} = 100$ . So, we get,
$\Rightarrow \log 200 = \log \left( 2 \right) + 2$
Now, substituting the value of $\log 2$ given in the question in the expression, we get,
$\Rightarrow \log 200 = 0.3010 + 2$
Simplifying the expression further, we get,
$\Rightarrow \log 200 = 2.3010$
So, the correct answer is “2.3010”.

Note : There are several laws of the logarithm that make the calculations easier and help us evaluate the logarithm functions. The standard base of logarithm functions is $10$ , that is, if we are given a function without any base like $\log x$ then we take the base as $10$ . Now while applying the laws of the logarithm, we should keep in mind an important rule that is the base of the logarithm functions involved should be the same in all the calculations.