Question

Solve the following expression: \[\log 6+2\log 5+\log 4-\log 3-\log 2\]

Answer 1

Hint: We use the formula \[{{\log }_{b}}(mn)={{\log }_{b}}m+{{\log }_{b}}n\] to solve this question. We will begin with breaking log 6 and log 4 into simplest numbers using this formula. And then we will see if any similar terms cancel or any addition of similar terms is taking place.

Complete step-by-step answer:
Before proceeding with the question, we must know that logarithm has various important properties that are multiplication and division of logarithms which can be written in the form of logarithm of addition and subtraction.
There are three basic rules of logarithms as given below:
The first rule is \[{{\log }_{b}}(mn)={{\log }_{b}}m+{{\log }_{b}}n......(1)\]. In this rule, the multiplication of two logarithmic values is equal to the addition of their individual logarithms.
The second rule is \[{{\log }_{b}}\left( \dfrac{m}{n} \right)={{\log }_{b}}m-{{\log }_{b}}n........(2)\]. This is called the division rule. Here the division of two logarithmic values is equal to the difference of each logarithm.
The third rule is \[{{\log }_{b}}({{m}^{n}})=n{{\log }_{b}}m........(3)\]. This is the exponential rule of logarithms. The logarithm of m with a rational exponent is equal to the exponent times its logarithm.
Using these rules we can write,
 \[\begin{align}
  & \,\Rightarrow \log 6+2\log 5+\log 4-\log 3-\log 2 \\
 & \,\Rightarrow \,\log (2\times 3)+2\log 5+\log (2\times 2)-\log 3-\log 2.....(4) \\
\end{align}\]
Applying rule 1 in equation (4) we get,
 \[\,\Rightarrow \log 2+\log 3+2\log 5+\log 2+\log 2-\log 3-\log 2.....(5)\]
Rearranging terms and simplifying equation (5) we get,
 \[\,\Rightarrow 2\log 2+2\log 5....(6)\]
Taking 2 common in equation (6) we get,
 \[\,\Rightarrow 2(\log 2+\log 5).......(7)\]
Applying rule 1 again in equation (7) we get,
 \[\,\Rightarrow 2\log (5\times 2)=2\log 10=2\]
Hence 2 is the answer to this question as \[{{\log }_{10}}10=1\].

Note: We have to be thorough with the basic rules of logarithm and apply it to break any larger terms into simpler terms. Students should note that whenever the base of logarithm is not given then it is assumed that base is 10.