Question

Expand the following:
$\log \left( {\dfrac{{{p^2}{q^3}}}{{{r^4}}}} \right)$

Answer 1

Hint : To solve this problem, we first need to consider all the mathematical relations involved in the given term which needs to be expanded. After that, we will apply the rules of logarithms to expand it. We need to use the multiplication rule, division rule and exponential rule of logarithms.
Formulas used:
$\log (ab) = \log a + \log b$
$\log \left( {\dfrac{a}{b}} \right) = \log a - \log b$
$\log \left( {{a^b}} \right) = b\log a$

Complete step-by-step answer :
We are asked to expand the term $\log \left( {\dfrac{{{p^2}{q^3}}}{{{r^4}}}} \right)$ .
Here, first we will apply the division rule $\log \left( {\dfrac{a}{b}} \right) = \log a - \log b$ in which we will take $a = {p^2}{q^3}$ and $b = {r^4}$ .
$\Rightarrow \log \left( {\dfrac{{{p^2}{q^3}}}{{{r^4}}}} \right) = \log \left( {{p^2}{q^3}} \right) - \log {r^4}$
Now we will consider the term $\log \left( {{p^2}{q^3}} \right)$ and apply the multiplication rule $\log (ab) = \log a + \log b$ for this term. For that, we will take $a = {p^2}$ and $b = {q^3}$ .
$\Rightarrow \log \left( {{p^2}{q^3}} \right) = \log {p^2} + \log {q^3}$
If we put this in the expansion of the main term we get
$\Rightarrow \log \left( {\dfrac{{{p^2}{q^3}}}{{{r^4}}}} \right) = \log {p^2} + \log {q^3} - \log {r^4}$
Now, we will consider all the three terms $\log {p^2}$ , $\log {q^3}$ and $\log {r^4}$ and apply exponential rules to them respectively.
For the term $\log {p^2}$ , if we apply the exponential rule $\log \left( {{a^b}} \right) = b\log a$ by taking $a = p$ and $b = 2$ , we get $\log {p^2} = 2\log p$ .
For the term $\log {q^3}$ , if we apply the exponential rule $\log \left( {{a^b}} \right) = b\log a$ by taking $a = q$ and $b = 3$ , we get $\log {q^3} = 3\log q$ .
For the term $\log {r^4}$ , if we apply the exponential rule $\log \left( {{a^b}} \right) = b\log a$ by taking $a = r$ and $b = 4$ , we get $\log {r^4} = 4\log r$ .
Putting all these values in the expansion of the main term, we get
$\Rightarrow \log \left( {\dfrac{{{p^2}{q^3}}}{{{r^4}}}} \right) = 2\log p + 3\log q - 4\log r$
Thus, our final answer is $2\log p + 3\log q - 4\log r$
So, the correct answer is “$2\log p + 3\log q - 4\log r$”.

Note : For solving this problem, we have used three rules of logarithms. First, we have used the division rule, which states that the division of two logarithmic values is equal to the difference of each logarithm. After that, we have used the multiplication rule for logarithms which states that the multiplication of two logarithmic values is equal to the addition of their individual logarithms. Finally, we have used the exponential rule for logarithms which states that the logarithm of any term with a rational exponent is equal to the exponent times its logarithm.