Question

How do you simplify $2\log 3 + \log 4 - \log 6$ ?

Answer 1

Hint: In this question, we have been asked how do we simplify the given expression in terms of logarithmic. Along with solving the question, we have to mention the steps as well. You must be aware about all the logarithmic properties. First, simplify the first term. Then, use the logarithmic property of addition to simplify the first two terms. You will get two terms to simplify. Use the logarithmic property of subtraction to simplify these two terms. You will get your answer.

Formula used: 1) $m\log n = \log {n^m}$
2) $\log m + \log n = \log mn$
3) $\log m - \log n = \log \dfrac{m}{n}$

Complete step-by-step solution:
We are given an expression in terms of logarithm and we have been asked to simplify it.
Step 1: At first, we will simplify the first term using the identity $m\log n = \log {n^m}$.
$ \Rightarrow 2\log 3$ …. (First term of the given expression)
Using the identity,
$ \Rightarrow 2\log 3 = \log {3^2} = \log 9$
Putting this back in the given expression,
$ \Rightarrow \log 9 + \log 4 - \log 6$
Step 2: In this step, we will use the identity $\log m + \log n = \log mn$ in the first two terms,
$ \Rightarrow \log \left( {9 \times 4} \right) - \log 6$
On simplifying, we will get,
$ \Rightarrow \log 36 - \log 6$
Step 3: Now, in the last step, we will use the logarithmic property $\log m - \log n = \log \dfrac{m}{n}$ to finally get our answer.
$ \Rightarrow \log \dfrac{{36}}{6}$
On simplifying this, we will get,
$ \Rightarrow \log 6$

Hence, $2\log 3 + \log 4 - \log 6 = \log 6$.

Note: There are various other properties of log which we should be aware about. These are –
1) Base conversion formula - ${\log _a}b = \dfrac{{\log b}}{{\log a}}$.
This method is used to convert the base. This is a very important formula as we cannot do a question further if the base of every term is not the same.
2) ${\log _a}b = \dfrac{1}{{{{\log }_b}a}}$
3) ${\log _{{a^m}}}{b^n} = \dfrac{{n\log b}}{{m\log a}}$
4) $\log 1 = 0$
5) Quotient rule – $\log m - \log n = \log \dfrac{m}{n}$
6) Product rule – $\log m + \log n = \log mn$