Solving for x in logarithm problem

  • Thread starter Sumedh
  • Start date
  • #1
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Homework Statement


The equation (attached as image) has
(a)one irrational solution
(b)no prime solution
(c)two real solutions
(d)one integral solution

i would like to get help on how to find the possible values of x

Homework Equations


( the equation is attached below)


The Attempt at a Solution


i solved the equation by applying the base changing property and then doing
L.C.M of both equations but after that i am unable to solve further?

Any help will be highly appreciated.
 

Attachments

  • equation.png
    equation.png
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Answers and Replies

  • #2
SammyS
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Show what you tried so far.

What results did you have?

I suggest: Do change of base with base of 2 .
 
  • #3
62
0
The equation (attached as image) has
(a)one irrational solution
(b)no prime solution
(c)two real solutions
(d)one integral solution

i would like to get help on how to find the possible values of x

i solved the equation by applying the base changing property and then doing
L.C.M of both equations but after that i am unable to solve further?

Any help will be highly appreciated.
 

Attachments

  • equatin.png
    equatin.png
    920 bytes · Views: 432
  • #4
218
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Because the numbers seem to be related to powers of 2, I was convinced to work with logarithm base 2. Let [itex]lb[/itex] stand for [itex]log_{2}[/itex].

We then have

[tex]log_{x^2}(16) + log_{2x}(64)[/tex]
[tex]= \frac{lb(16)}{2\cdot lb(x)} + \frac{lb(64)}{lb(2x)}[/tex]
[tex]= \frac{2}{lb(x)} + \frac{6}{lb(x) + 1} = 3[/tex]

this reduces to a quadratic which should be easily solvable.
 
  • #5
62
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2/log2(x) + 6/ (log2(x)+1)=3
i got this result
after that what should i do
 
  • #6
209
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logx^2 16 + log2x 64 = 3
log(16) /log (x2) + log(64) /log(2x) = 3
log(2x) log(16) + log(64) log (x2) = 3 log(2x) log (x2)
log(2x) log(24) + log(26) log (x2) = 3 log(2x) log (x2)
4 (log 2+log x) log(2) + 12 log(2) log (x) = 6 (log 2 + log x) log x
4 (log 2)2 + 4 log (x) log (2) + 12 log (2) log (x) = 6 log (2) log (x) + 6 (log x)2
4 (log 2)2 + 10 log (2) log (x) = 6 (log x)2

Solve the quadratic formula for log x and you'll get your answer.
 
Last edited:
  • #7
62
0
:smile:Thanks a lot i have got the answer:smile:
answer are x=4 and x=1[itex]/\sqrt[3]{}2[/itex]
 

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