Problem:
A line x=k intersects the graph of y=log5x and the graph of y=log5(x+4). The distance between the points of intersection is 0.5. Given that k=a+b, where a and b are integers, what is a+b?
Answer Choices:
A. 6
B. 7
C. 8
D. 9
E. 10
Solution:
The line x=k intersects y=log5(x+4) and y=log5x at (k,log5(k+4)) and (k,log5k), respectively. Since the length of the vertical segment is 0.5
0.5=log5(k+4)−log5k=log5kk+4
so kk+4=5. Solving for k yields k=5−14=1+5, so a+b=6.