If $\log \left( {1 + x} \right) + \log \left( {x - 1} \right) = \log 8$ , then $x$ is equal to
$
{\text{A}}{\text{. 2}} \\
{\text{B}}{\text{. 3}} \\
{\text{C}}{\text{. - 3}} \\
{\text{D}}{\text{. - 2}} \\
$
Last updated date: 19th Mar 2023
•
Total views: 305.1k
•
Views today: 7.84k
Answer
305.1k+ views
Hint: In this question we have to find the value of $x$, so the key concept is to apply the basic logarithmic identities in the given equation $\log \left( {1 + x} \right) + \log \left( {x - 1} \right) = \log 8$ to get the correct value of $x$.
Complete step-by-step answer:
We have been given that $\log \left( {1 + x} \right) + \log \left( {x - 1} \right) = \log 8$ …………. (1)
We know that, if $a > 0,b > 0$ then we have,
$ \Rightarrow \log a + \log b = \log ab$
So, equation (1) can also be written as
$
\Rightarrow \log (1 + x) + \log (x - 1) = \log 8 \\
\Rightarrow \log \left\{ {\left( {1 + x} \right)(x - 1)} \right\} = \log 8 \\
$ ………… (2)
Now we can write $\left( {1 + x} \right)\left( {x - 1} \right) = {x^2} - {1^2} = {x^2} - 1$
So, equation (2) will become
$ \Rightarrow \log ({x^2} - 1) = \log 8$ ………….. (3)
We also know that if $\log a = \log b$ then we have,
$ \Rightarrow a = b$ for all $a > 0,b > 0$
So, equation (3) will become
$
\Rightarrow \log ({x^2} - 1) = \log 8 \\
\Rightarrow {x^2} - 1 = 8 \\
\Rightarrow {x^2} = 9 \\
\Rightarrow x = \pm \sqrt 9 \\
\Rightarrow x = \pm 3 \\
\Rightarrow x = + 3, - 3 \\
$
Here we get the two values of $x = + 3, - 3$
And we have the equation (1) is $\log \left( {1 + x} \right) + \log \left( {x - 1} \right) = \log 8$
So, for $x = - 3$ equation (1) is not defined because ‘$\log $’ is only defined for positive real numbers.
But for $x = + 3$ equation (1) is defined.
Hence option B is the correct answer.
Note: Whenever we face such types of problems the key point is that to simplify the problems by using basic logarithmic identities. So we have always remembered the basic logarithmic identities. After getting the solutions the most important step is rechecking whether the given equation will define or not for our founded solutions. The solution for which the given equation is defined is our right answer.
Complete step-by-step answer:
We have been given that $\log \left( {1 + x} \right) + \log \left( {x - 1} \right) = \log 8$ …………. (1)
We know that, if $a > 0,b > 0$ then we have,
$ \Rightarrow \log a + \log b = \log ab$
So, equation (1) can also be written as
$
\Rightarrow \log (1 + x) + \log (x - 1) = \log 8 \\
\Rightarrow \log \left\{ {\left( {1 + x} \right)(x - 1)} \right\} = \log 8 \\
$ ………… (2)
Now we can write $\left( {1 + x} \right)\left( {x - 1} \right) = {x^2} - {1^2} = {x^2} - 1$
So, equation (2) will become
$ \Rightarrow \log ({x^2} - 1) = \log 8$ ………….. (3)
We also know that if $\log a = \log b$ then we have,
$ \Rightarrow a = b$ for all $a > 0,b > 0$
So, equation (3) will become
$
\Rightarrow \log ({x^2} - 1) = \log 8 \\
\Rightarrow {x^2} - 1 = 8 \\
\Rightarrow {x^2} = 9 \\
\Rightarrow x = \pm \sqrt 9 \\
\Rightarrow x = \pm 3 \\
\Rightarrow x = + 3, - 3 \\
$
Here we get the two values of $x = + 3, - 3$
And we have the equation (1) is $\log \left( {1 + x} \right) + \log \left( {x - 1} \right) = \log 8$
So, for $x = - 3$ equation (1) is not defined because ‘$\log $’ is only defined for positive real numbers.
But for $x = + 3$ equation (1) is defined.
Hence option B is the correct answer.
Note: Whenever we face such types of problems the key point is that to simplify the problems by using basic logarithmic identities. So we have always remembered the basic logarithmic identities. After getting the solutions the most important step is rechecking whether the given equation will define or not for our founded solutions. The solution for which the given equation is defined is our right answer.
Recently Updated Pages
Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts
Name the Largest and the Smallest Cell in the Human Body ?

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

A ball impinges directly on a similar ball at rest class 11 physics CBSE

Lysosomes are known as suicidal bags of cell why class 11 biology CBSE

Two balls are dropped from different heights at different class 11 physics CBSE

A 30 solution of H2O2 is marketed as 100 volume hydrogen class 11 chemistry JEE_Main

A sample of an ideal gas is expanded from 1dm3 to 3dm3 class 11 chemistry CBSE
