Question

How do you expand \[\log {{\left( \dfrac{6}{11} \right)}^{5}}\]?

Answer 1

Hint: This type of problem is based on the concept of logarithm. First, we have to consider the given function. Using the property of logarithm, that is \[\log {{a}^{n}}=n\log a\], we can simplify the given expression. Here, we find that n=5. Now, we have to use the property of logarithm, that is, \[\log \left( \dfrac{a}{b} \right)=\log a-\log b\] to expand further. Here, by comparing with the given question, we get a=6 and b=11. Then, we need to use the multiplication property of logarithm, that is, \[\log \left( ab \right)=\log a+\log b\] in log6 since 6 can be written as the product of 2 and 3. Thus, we get the final required answer.

Complete step by step solution:
According to the question, we are asked to expand the given function \[\log {{\left( \dfrac{6}{11} \right)}^{5}}\].
We have been given the function is \[\log {{\left( \dfrac{6}{11} \right)}^{5}}\]. -------------(1)
We know that \[\log {{a}^{n}}=n\log a\].
Let us use this property of logarithm in the function (1).
Here, we find that n=5.
Therefore, we get
\[\Rightarrow \log {{\left( \dfrac{6}{11} \right)}^{5}}=5\log \left( \dfrac{6}{11} \right)\]
Now, we have to expand the function further.
We know that \[\log \left( \dfrac{a}{b} \right)=\log a-\log b\].
Let us use this property of logarithm in the above function.
Here, we find that a=6 and b=11.
Therefore, we get
\[\log {{\left( \dfrac{6}{11} \right)}^{5}}=5\left[ \log 6-\log 11 \right]\].-----------(2)
Here, we find that 11 is a prime number.
But 6 can be written as a product of 3 and 2.
\[\Rightarrow \log {{\left( \dfrac{6}{11} \right)}^{5}}=5\left[ \log \left( 2\times 3 \right)-\log 11 \right]\]
We can use the multiplication rule of logarithm, that is \[\log \left( ab \right)=\log a+\log b\], in log 6.
Hence, we can write log6 as
log6=log2+log3.
Let us substitute the above result in equation (2).
We get
\[\log {{\left( \dfrac{6}{11} \right)}^{5}}=5\left[ \log 2+\log 3-\log 11 \right]\]
We cannot expand the function further.
Therefore, the expansion of the function \[\log {{\left( \dfrac{6}{11} \right)}^{5}}\] is \[5\left[ \log 2+\log 3-\log 11 \right]\].

Note: For this type of problems, we can expand the given function until we get a prime number in each logarithm. We should be thorough with the properties of logarithm and should not get confused with the multiplication and division rule of logarithm. Also avoid calculation mistakes based on sign conventions.