Question

Logx - logx-1^2=2log(x-1)

Answer 1

Answer:

x is approximately 2.220744

Step-by-step explanation:

This can be simplified a little using properties of logarithms, and then solve it by graphing:

$log(x)-log(x-1)^2=2\,log(x-1)\\log(x)-2\,log(x-1)=2\,log(x-1)\\log(x)=4\,log(x-1)$

So we use a graphing tool to find the intersection point of the graph of $log(x)$, and the graph of $4\,log(x-1)$

Please see attached image for the graph and solution.

The value of x is approximately 2.220744

Answer 2

Answer:

  x = 2.32011574011

Step-by-step explanation:

The problem with your original equation is that it is a long way of saying ...

  log(x) -log(x) -1 = 2log(x-1)

  0 -1 = 2log(x-1)

which has the solution ...

  -1/2 = log(x -1)

  1/√10 = x -1

  x = 1 + 1/√10 ≈ 1.3162278

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We have asked for clarification, and what we got was ...

  $\log{(x)}-\log{(x-1^2)}=2\log{(x-1)}$

which, again, is a long way of saying ...

  $\log{(x)}-\log{(x-1)}=2\log{(x-1)}$

The other reasonable interpretation of your 'clarified' equation is ...

  $\log{(x)}-\log{((x-1)^2)}=2\log{(x-1)}$

which you already have an answer to. You have declared that a "misconception."

So, we are left with the interpretation that the equation you want a solution to is ...

  $\log{(x)}-\log{(x-1)}=2\log{(x-1)}$

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When solving these graphically, I like to write the equation as a function whose zero(s) we're trying to find. For this, when we subtract the right side, we get ...

  $f(x)=\log{(x)}-3\log{(x-1)}$

A graphing calculator shows that f(x) = 0 when ...

  x ≈ 2.32011574011

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If you don't like my interpretation, check out the second attachment. It has your x-1² as the argument of the middle term. You can see that the calculator interpreted that the same way I did (as required by the order of operations).