Question

Find the domain of this problem, do not find the solution I have the solution it is x=2, I need the domain.

Answer 1

Answer:

x > 1/2

Step-by-step explanation:

The lower limint of the domain of the log function, regardless of the base, is x > 0.  But that's not all.  If the right side is 2, then there's an upper limit to x.

If the solution is x = 2, then we have:

log             4.5x = 2

      2(2)-1

or...

log     9 = 2.  This is a true statement, since 9 = 3^2.

      3

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The question here is whether or not there's an upper limit on x.

You claim that the solution is x=2.  Is that the upper limit of the domain?  Let's find out.  Let's try x = 3:

log       13.5

      5               = 2  iS THIS TRUE OR FALSE?

Use the change of base formula:

log     13.5 = log 13.5 / log 5 = 1.617.  This is actually less than 2

      5.

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Let me play the devil's advocate and hypothesize for a moment that the solution is not x = 2.  If not 2, then what is x?

(2x - 1) ^ log           (4.5x)     =    (2x -1)^2, or 4x^2 - 4x + 1

                    2x -1

Then 4.5x = 4x^2 - 4x + 1, or

4x^2 - 8.5x + 1 = 0

This is a quadratic function, with a = 4, b = - 8.5 and c = 1.  Then the discriminant is b^2 - 4(a)(c), which here is (-8.5)^2 - 4(4)(1) ), or 56.25.

This would lead to roots of

      - (-8.5) plus or minus √56.25

x = ---------------------------------------------------          

                            2(4)

x = 2 or x = 1/8    (You claim that the solution is x = 2).

Let's try once more to identify the domain:

(1) log          (4.5x) = 2  is valid only when the base, 2x-1, is greater than                

          2x-1               zero.  Setting 2x - 1 > 0, we get 2x ≥ 1. or x > 1/2.  For values of x less than 1/2, the base (2x - 1) would be 0 or smaller, which is not permissible when working with logs.

This domain, x > 1/2, applies only to the function log          y

                                                                                        2x-1

You claim that x = 2.  Does that satisfy x > 1/2, the domain of x?  Yes.  So that's promising.

Based upon these explorations, I'd claim that the domain of the log part of this equation is x > 1/2.