Answer

Verified

408.9k+ views

**Hint:**Apply the formula: - \[\log m-\log n=\log \left( \dfrac{m}{n} \right)\] to simplify the L.H.S. Now, remove the log function from both the sides and cross – multiply the terms to form a quadratic equation in x. Solve this quadratic equation with the help of the discriminant formula: - \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]. Here, a = coefficient of \[{{x}^{2}}\], b = coefficient of x and c = constant term. Reject the value of x that is invalid by using the information that the” argument of the log must be greater than 0”.

**Complete step-by-step solution:**

Here, we have been provided with the logarithmic equation: \[\ln \left( x+1 \right)-\ln \left( x-2 \right)=\ln x\] and we have been asked to solve it. That means we have to find the value of x.

Now, using the formula: - \[\log m-\log n=\log \left( \dfrac{m}{n} \right)\] in the L.H.S., we get,

\[\Rightarrow \ln \left( \dfrac{x+1}{x-2} \right)=\ln x\]

Comparing the argument of log on both the sides by removing the logarithmic function, we get,

\[\Rightarrow \dfrac{x+1}{x-2}=x\]

Cross – multiplying the terms, we get,

\[\begin{align}

& \Rightarrow x+1=x\left( x-2 \right) \\

& \Rightarrow x+1={{x}^{2}}-2x \\

& \Rightarrow {{x}^{2}}-3x-1=0 \\

\end{align}\]

Assuming the coefficient of \[{{x}^{2}}\], coefficient of x and constant term as a, b and c respectively, we have,

\[\Rightarrow \] a = 1, b = -3, c = -1

Applying the discriminant formula given as: - \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\], we get,

\[\Rightarrow x=\dfrac{-\left( -3 \right)\pm \sqrt{{{\left( -3 \right)}^{2}}-4\left( 1 \right)\left( -1 \right)}}{2\times 1}\]

\[\Rightarrow x=\dfrac{3\pm \sqrt{13}}{2}\]

\[\Rightarrow x=\dfrac{3+\sqrt{13}}{2}\] or \[x=\dfrac{3-\sqrt{13}}{2}\]

Here, we have obtained two values of x. Now, let us check if any of the two values in invalid or not.

We know that a logarithmic function is only defined when its argument and base is greater than 0 and base is unequal to 1. In the above question we have ln, i.e., log to the base e, where the value of e is nearly 2.71 which is greater than 0. So, the base is defined. Now, let us define the argument. We have x, x + 1 and x – 2 as the arguments. So, we must have,

(i) \[x>0\Rightarrow x\in \left( 0,\infty \right)\]

(ii) \[x+1>0\Rightarrow x>-1\Rightarrow x\in \left( -1,\infty \right)\]

(iii) \[x-2>0\Rightarrow x>2\Rightarrow x\in \left( 2,\infty \right)\]

Since, we need to satisfy all the three conditions, therefore we must consider the intersection of the three sets of values of x obtained. So, we have,

Therefore, \[x\in \left( 2,\infty \right)\] is the final condition.

Now, we can see that \[\dfrac{3-\sqrt{13}}{2}\] will be negative because \[\sqrt{13}\] is greater than 3, so \[x=\dfrac{3-\sqrt{13}}{2}\] does not satisfy the above condition. So, it must be rejected.

**Hence, \[x=\dfrac{3+\sqrt{13}}{2}\] will be our answer.**

**Note:**One may note that we cannot remove the log function directly from the initial expression: - \[\ln \left( x+1 \right)-\ln \left( x-2 \right)=\ln x\] as it will be a wrong approach. First, we need to convert the two logarithmic terms in the L.H.S. into a single logarithmic term in the L.H.S. into a single logarithmic term by using the difference to quotient rule and then only we can remove the function. Remember that we do not just have to calculate the value of x but we must check if it satisfies the domain or not. We must reject the invalid value as it makes the function undefined.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Choose the antonym of the word given below Furious class 9 english CBSE

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Change the following sentences into negative and interrogative class 10 english CBSE