Every point (x,y) on the curve y=log23x is transferred to a new point by the following translation (x′,y′)=(x+m,y+n), where m and n are integers. The set of (x′,y′) form the curve y=log2(12x−96). What is the value of m+n?
To solve this problem, we need to find the value of m and n using the given information.
We have the equation y = log2(3x) for the curve. Let's consider an arbitrary point on this curve, (x, y).
According to the translation, the new point (x', y') is obtained by adding m to x and adding n to y. Mathematically, we can represent this as:
x' = x + m
y' = y + n
We are also given the equation for the new curve y = log2(12x - 96). We need to find the relationship between (x', y') and x in order to determine the values of m and n.
To do this, we substitute x' = x + m into the equation for the new curve. This gives:
y = log2(12(x + m) - 96)
Simplifying further:
y = log2(12x + 12m - 96)
Comparing this with y' = log2(12x - 96), we can equate the expressions on both sides:
log2(12x + 12m - 96) = log2(12x - 96)
Since both sides have the same base (log2), the arguments of the logarithms must be equal:
12x + 12m - 96 = 12x - 96
Now, we can cancel out the 12x terms on both sides:
12m - 96 = 0
Solving for m:
12m = 96
m = 8
Now that we have the value of m, we can substitute it back into the equation for x' = x + m to find x':
x' = x + 8
Since the translation only affects x and not y, n must be zero (n = 0).
Therefore, the value of m + n is:
m + n = 8 + 0 = 8