Question

How do you condense $ \ln \left( 8 \right) - \ln \left( x \right) $ ?

Answer 1

Hint: In the above question, we are given a difference of two functions and we have to find their value, the given functions are logarithmic functions. So to solve this question, 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 this question, we will use the laws of the logarithm.

Complete step-by-step answer:
We have to condense $ \ln \left( 8 \right) - \ln \left( x \right) $ .
We know that
 $ {\log _a}x - {\log _a}y = {\log _a}\left( {\dfrac{x}{y}} \right) $
Hence, $ \ln \left( 8 \right) - \ln \left( x \right) $ can also be condensed in the same way using the logarithmic identity.
 $ \ln \left( 8 \right) - \ln \left( x \right) = \ln \left( {\dfrac{8}{x}} \right) $
So, the correct answer is “ $ \ln \left( {\dfrac{8}{x}} \right) $ ”.

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. But in the given question, we are given the function $ \ln x $ . $ \ln x $ is a logarithmic function with base of $ \log $ as e. So, the base of both $ \ln 8 $ and $ \ln x $ is e. 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, as the base of both the functions in the question is the same, we can apply the logarithm laws in the given question.