Expert: Paul Klarreich - 9/18/2007

Question
Hi Paul,
My Q :

A line x= k intersects the graph of y = log5 x 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
Questioner:   pnd
Category:  Advanced Math
Private:  No
 
Subject:  Graphs
Question:  Hi Paul,
My Q :

A line x= k intersects the graph of y = log5 x 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 ?
............................................
Hi, Pnd,

[I assume that when you wrote:  log5 x, you meant 'the base-5 logarithm of x'.]

Your line  x = k  is vertical, so it passes through a point on each graph for the same value of x, i.e.  x = k.  That means:

y1 = log5(k),  and
y2 = log5(k + 4), and
y2 - y1 = 0.5 = 1/2

So we have an equation:

log5(k + 4) - log5(k) = 1/2

and we use some log properties:
     k + 4
log5(-------) = 1/2
       k

k + 4
------- = 5^1/2 = sqrt(5)
  k

Solve for k:

k + 4 = k sqrt(5)

4 = k sqrt(5) - k
        4
k = -----------
   sqrt(5) - 1

This is not an integer; it's about  3.236

So the next part of the problem does not make any sense.  Perhaps you left something out or your fingers slipped.  Why don't you check it out and, perhaps, resubmit it?  [Of course, the above calculation might be enough to give you the clue about how to do it anyway.]