Question

How do you use the logarithmic property to expand \[{{\log }_{10}}\left( \dfrac{x{{y}^{4}}}{{{z}^{3}}} \right)\] ?

Answer 1

Hint: In the above stated question we can clearly see that in the above question we will need the basic logarithmic property; this property of logarithmic will help us expand any logarithmic function. The property that we will use is the product, quotient and power property of logarithm, with the help of these three properties we will be able to expand the logarithm function that is mentioned in the question.

Complete step by step answer:
In the above stated question we are given with the logarithmic function which needs to be expanded from the form mentioned in the question. For doing this we will use basic logarithmic properties which are product, quotient and power property of logarithm. Let us take a logarithmic function as: \[{{\log }_{a}}b=c\]
In this above logarithmic function “a” is the base of the function “b” is the logarithmic function value which needs to be solved with the help of logarithm and “c” is the value of the logarithm when it is solved. Now to solve the above question “b” is a mixture of many different variables and to solve those we will be needing three different properties which are as mentioned above first being quotient property to convert the fraction into normalized form with which we will get:
\[{{\log }_{10}}x{{y}^{4}}-{{\log }_{10}}{{z}^{3}}\]
Now we will use the product property of logarithm in the first part to different between x and y and we will get:
\[{{\log }_{10}}x+{{\log }_{10}}{{y}^{4}}-{{\log }_{10}}{{z}^{3}}\]
Now we will be using the power property of logarithm to remove the powers from y and z variables and we will get:
\[{{\log }_{10}}x+4{{\log }_{10}}y-3{{\log }_{10}}z\]
So we get the expanded form of the whole logarithmic equation to be \[{{\log }_{10}}x+4{{\log }_{10}}y-3{{\log }_{10}}z\].

Note: In the above stated question we can see that how the basics of logarithmic function can come in handy so try to remember the basic logarithmic function which is logarithm of product, logarithm of quotient and logarithm of power.