Question

Simplify the given logarithmic functions: log (ab) – log |b|
(A). log a
(B). log |a|
(C). -log a
(D). None of these

Answer 1

Hint: Start by simplifying the relation by using the logarithmic properties and rearrange the like terms . Then look for all the possible cases of |b| and solve by substituting those values in the equation, We’ll get the answer.

Complete step-by-step answer:
Given
$\log (ab)-\log \left|b\right|$
By using the property of logarithmic functions i.e. $\log x-\log y=\log {\displaystyle \frac{x}{y}}$ . We get , $\log {\displaystyle \frac{ab}{\left|b\right|}}$
Now, The equation can also be written as
$\log a\cdot {\displaystyle \frac{b}{\left|b\right|}}\to eqn(1)$
Now, We’ll solve for all the possible cases for ${\displaystyle \frac{b}{\left|b\right|}}$
Case 1: $\left|b\right|=b$
Then , the value of ${\displaystyle \frac{b}{\left|b\right|}}$ will be +1
Case 2: $\left|b\right|=-b$
 Then , the value of ${\displaystyle \frac{b}{\left|b\right|}}$ will be ${\displaystyle \frac{b}{-b}}=-1$
Now ,Equation 1 can be written as
 $\log a\cdot (1)\text{ and }\log a\cdot (-1)$
Which gives us two values $\log a\text{ and }\log (-a)$ , and can be represented as $\log \left|a\right|$
Therefore, the answer is $\log \left|a\right|$.
So , option B is the correct answer.

Note:All the properties of logarithmic functions must be well known and practised in order to solve such similar questions. Some other functions such as the G.I.F.( Greatest Integer Function) and Modulus function etc must also be known , as a combination of such functions can be asked.