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Question: How do you solve \({\log _2}\left( {x - 6} \right) + {\log _2}\left( {x - 4} \right) = {\log _2}x\)....

How do you solve log⁡2(x−6)+log⁡2(x−4)=log⁡2x{\log _2}\left( {x - 6} \right) + {\log _2}\left( {x - 4} \right) = {\log _2}x.

Explanation

Solution

Given a logarithmic expression. We have to find the value of the expression. First, we will apply the logarithmic property to the left-hand side of the expression. Then, write the equation in the form of the quadratic equation. Then, we will find the factors of the equation. Then, we will find the solution to the equation by setting each factor equal to zero.

Formula used:
The multiplication property of the logarithms is given by:
log⁡xa+log⁡xb=log⁡xab{\log _x}a + {\log _x}b = {\log _x}ab
The law of logarithms is given by:
If log⁡xa=log⁡xb{\log _x}a = {\log _x}b, then a=ba = b

Complete step-by-step answer:
We are given the expression, log⁡2(x−6)+log⁡2(x−4)=log⁡2x{\log _2}\left( {x - 6} \right) + {\log _2}\left( {x - 4} \right) = {\log _2}x. First, we will apply the multiplication property of the logarithms to the left hand side of the expression.
⇒log⁡2[(x−6)(x−4)]=log⁡2x\Rightarrow {\log _2}\left[ {\left( {x - 6} \right)\left( {x - 4} \right)} \right] = {\log _2}x
Now, we will apply the law of logarithms to remove the log from both sides of the equation.
⇒(x−6)(x−4)=x\Rightarrow \left( {x - 6} \right)\left( {x - 4} \right) = x
Now, we will the distributive property to multiply the terms on the left hand side of the expression.
⇒x(x)+x(−4)+(−6)x+(−6)(−4)=x\Rightarrow x\left( x \right) + x\left( { - 4} \right) + \left( { - 6} \right)x + \left( { - 6} \right)\left( { - 4} \right) = x
On solving the expression further, we get:
⇒x2−4x−6x+24=x\Rightarrow {x^2} - 4x - 6x + 24 = x
Now, on combining like terms, we get:
⇒x2−10x+24=x\Rightarrow {x^2} - 10x + 24 = x
Now, subtract xx from both sides to write the equation in standard form.
⇒x2−10x+24−x=x−x\Rightarrow {x^2} - 10x + 24 - x = x - x
⇒x2−11x+24=0\Rightarrow {x^2} - 11x + 24 = 0
Now, factorize the equation by splitting the middle term whose sum is −11 - 11 and product is 2424.
⇒x2−8x−3x+24=0\Rightarrow {x^2} - 8x - 3x + 24 = 0
⇒x(x−8)−3(x−8)=0\Rightarrow x\left( {x - 8} \right) - 3\left( {x - 8} \right) = 0
⇒(x−8)(x−3)=0\Rightarrow \left( {x - 8} \right)\left( {x - 3} \right) = 0
Now, set each factor equal to zero to find the value of xx.
⇒(x−8)=0 or (x−3)=0\Rightarrow \left( {x - 8} \right) = 0{\text{ or }}\left( {x - 3} \right) = 0
⇒x=8 or x=3\Rightarrow x = 8{\text{ or }}x = 3

Final answer: Hence, the solution of the expression is x=3,8x = 3,8

Note:
In such types of questions the students mainly don't get an approach on how to solve it. In such types of questions students mainly forget to apply the correct law of logarithms. Students may get confused when to cancel out the log on both sides and equate the terms.