A student solved the equation below by graphing.
[tex]\[ \log _6(x-1)=\log _2(2x+2) \][/tex]

Which statement about the graph is true?

A. The curves do not intersect.
B. The curves intersect at one point.
C. The curves intersect at two points.
D. The curves appear to coincide.



Answer :

To determine the number of intersection points between the curves represented by the equations [tex]\(\log_6(x-1)\)[/tex] and [tex]\(\log_2(2x+2)\)[/tex], we need to solve the equation:

[tex]\[ \log_6(x-1) = \log_2(2x+2) \][/tex]

Let's break this down step-by-step.

1. Understand the Domain:
- [tex]\(\log_6(x-1)\)[/tex] is defined when [tex]\(x-1 > 0\)[/tex], hence:
[tex]\[ x > 1 \][/tex]
- [tex]\(\log_2(2x+2)\)[/tex] is defined when [tex]\(2x+2 > 0\)[/tex], hence:
[tex]\[ x > -1 \][/tex]
The intersection of these domains is [tex]\(x > 1\)[/tex].

2. Equate the Two Logs:
Rewriting the equation using the change of base formula:
[tex]\[ \frac{\log(x-1)}{\log(6)} = \frac{\log(2x+2)}{\log(2)} \][/tex]
This simplifies to:
[tex]\[ \log_2(2x+2) = \log_2(2(x+1)) \][/tex]

3. Equate the Arguments of the Logs:
Since [tex]\(\log_a(b) = \log_a(c)\)[/tex] if and only if [tex]\(b = c\)[/tex], we get:
[tex]\[ x-1 = 2^{\log_2(2(x+1))} \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ x-1 = 2(x+1) \][/tex]
Solving this equation:
[tex]\[ x - 1 = 2x + 2 \][/tex]
[tex]\[ -3 = x \][/tex]
This solution is not valid because [tex]\(-3\)[/tex] is not within the domain [tex]\(x > 1\)[/tex].

4. Look for Possible Other Solutions:
Let's check further by plotting or analyzing the behavior of both functions.

- [tex]\(\log_6(x-1)\)[/tex] increases logarithmically starting from [tex]\(x=1\)[/tex].
- [tex]\(\log_2(2x+2)\)[/tex] also increases logarithmically but grows faster than [tex]\(\log_6(x-1)\)[/tex] since the base of the logarithm [tex]\(2\)[/tex] is smaller than [tex]\(6\)[/tex].

Since the only algebraic solution (-3) does not lie in the valid domain [tex]\(x > 1\)[/tex], there are no valid solutions in the domain.

Thus, we conclude:
[tex]\[ \text{The curves do not intersect.} \][/tex]

Hence, the correct statement about the graph is:
- The curves do not intersect.