Answer

Verified

415.5k+ views

**Hint:**To solve the given question, first we apply the property of logarithm which states that if logs to the same base are added, then the numbers were multiplied, i.e. log (a) + log (b) = log (a.b). Then we simplify the equation further by using the definition of log, if log (a) = log (b) then a = b. and solve the equation in a way we solve the general quadratic equation.

**Formula used:**

The property of logarithm which states that if logs to the same base are added, then the

numbers were multiplied, i.e. log (a) + log (b) = log (a.b)

If log (a) = log (b) then a = b.

**Complete step by step solution:**

We have given that,

\[\ln \left( 4x-2 \right)-\ln 4=-\ln \left( x-2 \right)\]

Rearranging the terms in the above equation, we get

\[\Rightarrow \ln \left( 4x-2 \right)+\ln \left( x-2 \right)=\ln 4\]

Using the property of logarithm which states that if logs to the same base are added, then the numbers were multiplied, i.e. log (a) + log (b) = log (a.b)

Applying the above property, we get

\[\Rightarrow \ln \left( \left( 4x-2 \right)\times \left( x-2 \right) \right)=\ln 4\]

Using the definition of log, if log (a) = log (b) then a = b.

Applying the above property, we get

\[\Rightarrow \left( \left( 4x-2 \right)\times \left( x-2 \right) \right)=\ln 4\]

Simplifying the above equation, we get

\[\Rightarrow \left( 4x\times x \right)+\left( 4x\times -2 \right)+\left( -2\times x \right)+\left( -2\times -2

\right)=4\]

Simplifying further, we get

\[\Rightarrow 4{{x}^{2}}-8x-2x+4=4\]

\[\Rightarrow 4{{x}^{2}}-10x+4=4\]

Subtracting 4 from both the sides of the equation, we get

\[\Rightarrow 4{{x}^{2}}-10x=0\]

Taking out 2x as a common factor, we get

\[\Rightarrow 2x\left( 2x-5 \right)=0\]

Equation each factor equals to 0, we get

\[\Rightarrow 2x=0\] and \[2x-5=0\]

Now, solving

\[\Rightarrow 2x=0\]

\[\Rightarrow x=0\]

Now, solving

\[\Rightarrow 2x-5=0\]

Adding 5 to both the sides of the equation, we get

\[\Rightarrow 2x=5\]

Dividing both the sides of the equation by 2, we get

\[\Rightarrow x=\dfrac{5}{2}\]

Since, x > 2, so the only possible value of x is \[\dfrac{5}{2}\].

**Therefore, \[x=\dfrac{5}{2}\] is the required solution.**

**Note:**In the given question, we need to find the value of ‘x’. To solve these types of questions, we used the basic formulas of logarithm. Students should always require to keep in mind all the formulae for solving the question easily. After applying log formulae to the equation, we need to solve the equation in the way we solve general quadratic equations.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

A rainbow has circular shape because A The earth is class 11 physics CBSE

Which are the Top 10 Largest Countries of the World?

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

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

Give 10 examples for herbs , shrubs , climbers , creepers

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

Why is there a time difference of about 5 hours between class 10 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell