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The value of x, which satisfies 4x+6x=9x is
A. ln(51)ln2ln3ln2
B. ln(51)+ln2ln2ln3
C. ln(51)ln2ln2ln3
D. ln(51)+ln2ln3ln2

Answer
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Hint: In this question use the concept of logarithm and also remember the relation between common logarithm and natural logarithm i.e. log and ln which is given as; lnx=2.303logx or logx=lnx2.303 and use the formula log(ab)=logalogb to approach the solution.

Complete step-by-step answer:
According to the given information we have equation 4x+6x=9x
We have to find the value of x so now 4x+6x=9x (equation 1)
Now let’s divide equation (1) by 9x both the sides, we have
 (49)x+(69)x=9x9x
  (49)x+(23)x=1
We can write the above equation as

 ((23)2)x+(23)x=1 (equation 2)
Let us substitute (23)x=y in equation 2
 y2+y=1
  y2+y1=0
If we have a quadratic equation of the form ax2+bx+c=0
Then the roots of equation of this form is written as x= b±b24ac2a .
Now using the above to solve for the roots of equation y2+y1=0

 y= 1±52
So, we get 2 values of y that is y= 1+52 , y= 152
Now we have substituted y=(23)x , so let’s substitute it back
 y=(23)x=1±52
So
 (23)x=1±52
But (23)x=152 is not possible since (23)x will always be positive irrespective of value of x as exponential values aren’t negative.
So only value left with us is (23)x=1+52
Now take log both sides, so we get
 xlog(23)=log(512)
So, on simplifying
 x=log(512)log(23)
Now using the property of logarithm that log(ab)=logalogb
We have
 x=log(51)log2log2log3
As we know that lnx=2.303logx or logx=lnx2.303
Therefore, x=ln(51)2.303ln22.303ln22.303ln32.303
After simplifying the above equation, we get
x=ln(51)ln2ln2ln3
Therefore, the value of x is equal to ln(51)ln2ln2ln3

So, the correct answer is “Option C”.

Note: Whenever we face such types of problems the key step that we need we have in our mind is that we always try and form a quadratic equation by proper simplification and substitution. This quadratic equation thus formed can be easily solved (roots can be calculated) using the formula of quadratic formula for quadratic equations mentioned above.

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