Solve this please.---**Question Stem:**
7. 상수 a(a > 2)에 대하여 함수 y = log₂(x - a)의 그래프의 점근선이 두 곡선 y = log₂(x/4), y = log₁/₂ x와 만나는 점을 각각 A, B라 하자. AB = 4일 때, a의 값은? [3점]
**Translation of Question Stem:**
For a constant a (a > 2), let A and B be the points where the asymptote of the graph of the function y = log₂(x - a) intersects the two curves y = log₂(x/4) and y = log₁/₂ x, respectively. When the length of AB is 4, what is the value of a? [3 points]
**Mathematical Formulas/Equations Identified in Question Stem:**
* y = log₂(x - a)
* y = log₂(x/4)
* y = log₁/₂ x
* AB = 4 (length of line segment AB is 4)
**Options:**
① 4
② 6
③ 8
④ 10
⑤ 12
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We have three logarithmic functions to analyze. The first function y equals log base 2 of x minus a has a vertical asymptote at x equals a. This asymptote is a vertical line where the function is undefined. The other two functions are y equals log base 2 of x over 4 and y equals log base one half of x. Our goal is to find where the asymptote intersects these curves.
Now let's focus on the vertical asymptote of the function y equals log base 2 of x minus a. This asymptote occurs at x equals a because the logarithm is undefined when its argument equals zero, that is when x minus a equals zero. This vertical line x equals a will intersect our two curves at specific points. Point A is where the asymptote meets the curve y equals log base 2 of x over 4, and point B is where it meets the curve y equals log base one half of x.
Now let's systematically find the coordinates of points A and B. For point A, we solve where the line x equals a intersects the curve y equals log base 2 of x over 4. This gives us point A with coordinates a comma log base 2 of a over 4. We can simplify this using logarithm properties: log base 2 of a over 4 equals log base 2 of a minus log base 2 of 4, which equals log base 2 of a minus 2. For point B, we solve where x equals a intersects y equals log base one half of x. This gives us point B with coordinates a comma log base one half of a. Using the change of base property, log base one half of a equals negative log base 2 of a.
Now let's calculate the distance AB using the distance formula. Since both points A and B have the same x-coordinate x equals a, the distance AB is simply the absolute difference of their y-coordinates. We have AB equals the absolute value of y A minus y B, which is the absolute value of log base 2 of a over 4 minus log base one half of a. Substituting our simplified expressions, this becomes the absolute value of log base 2 of a minus 2, minus negative log base 2 of a, which simplifies to the absolute value of 2 log base 2 of a minus 2, or 2 times the absolute value of log base 2 of a minus 1. Since a is greater than 2, we have log base 2 of a greater than 1, so the absolute value becomes log base 2 of a minus 1. Therefore, AB equals 2 times log base 2 of a minus 1. Setting this equal to 4 gives us our equation to solve.
Now let's solve the equation 2 times log base 2 of a minus 1 equals 4 step by step. First, we divide both sides by 2 to get log base 2 of a minus 1 equals 2. Adding 1 to both sides gives us log base 2 of a equals 3. Therefore, a equals 2 to the power of 3, which is 8. Let's verify this solution by substituting back into our original problem. When a equals 8, point A has coordinates 8 comma log base 2 of 2, which is 8 comma 1. Point B has coordinates 8 comma log base one half of 8, which is 8 comma negative 3. The distance AB equals the absolute value of 1 minus negative 3, which equals 4. This confirms our answer. Therefore, the value of a is 8.