Question

What is the product rule of logarithms?

Answer 1

Hint : We first try to explain the product rule for the terms which are in logarithm. We use the formula of $\log a+\log b=\log \left( ab \right)$ . We take examples of constants to explain the process. Then we use variables to also show the use of the formula.

Complete step by step solution:
We first discuss the general form of the product rule of logarithms.
The product form of two numbers splits into a summation form of two separate logarithms.
The mathematical form will be $\log a+\log b=\log \left( ab \right)$ . We need to be careful about the base of the logarithm as that has to be the same for the formula to work.
Now we try to understand the formula with the use of examples.
Let’s assume we are adding the terms $\log 6$ and $\log 8$ .
Therefore, the simplified form will be the logarithm of the multiplication of 6 and 8.
This means $\log 6+\log 8=\log \left( 6\times 8 \right)=\log 48$ .
Similarly, if we want to break the $\log 6$ for $6=3\times 2$ , we will get
$\log 6=\log \left( 3\times 2 \right)=\log 3+\log 2$ .
We can do this for variables also.
Suppose we are given expression in the form of
${{\log }_{2}}\left( x+7 \right)+{{\log }_{2}}\left( x+8 \right)$
We operate the addition part in the left-hand side of
${{\log }_{2}}\left( x+7 \right)+{{\log }_{2}}\left( x+8 \right)$ .
{{\log }_{2}}\left( x+7 \right)+{{\log }_{2}}\left( x+8 \right)={{\log }_{2}}\left\\{ \left( x+7 \right)\left( x+8 \right) \right\\} .
The equation becomes {{\log }_{2}}\left\\{ \left( x+7 \right)\left( x+8 \right) \right\\}={{\log }_{2}}\left\\{ {{x}^{2}}+15x+56 \right\\} .

Note : We need to be careful about the base of the logarithm. Without base being the same we can’t operate the formula. For any ${{\log }_{b}}a$ , $a>0$ . This means for ${{\log }_{2}}\left( x+7 \right),{{\log }_{2}}\left( x+8 \right)$ , $\left( x+7 \right)>0$ and $\left( x+8 \right)>0$ . The simplified form is $x>-8$ .