Every point (x,y) on the curve y = \log_{2}{3x} 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 = \log_{2}{(12x-96)} . What is the value of m + n ?
To find the value of m + n, we need to understand the transformation that is applied to each point on the curve y = log₂(3x) to get the corresponding point on the curve y = log₂(12x - 96).
Let's start by considering a specific point (x, y) on the curve y = log₂(3x). When we apply the translation that moves this point to the corresponding point on the new curve, we get the new point (x', y') = (x + m, y + n).
We want to determine the values of m and n that give us the new curve y = log₂(12x - 96).
Comparing the equations of the two curves, we can see that the logarithms have the same base (base 2) and are equal to each other. Therefore, we can equate the expressions within the logarithms:
3x = 12x - 96
Solving this equation for x:
9x = 96
x = 96/9 = 32/3
Now that we have found the value of x, we can substitute it back into the equation of the original curve y = log₂(3x) to find the y-coordinate:
y = log₂(3 * 32/3) = log₂(32) = 5
So, the original point is (32/3, 5).
To find the corresponding point on the new curve, we apply the translation:
(x', y') = (32/3 + m, 5 + n)
Using the equation of the new curve y = log₂(12x - 96), we can plug in the x-coordinate and the y-coordinate:
5 + n = log₂(12 * (32/3) - 96)
5 + n = log₂(384 - 96)
5 + n = log₂(288)
To find the value of n, we need to solve this equation. However, logarithms are not directly solvable algebraically. We can use a calculator or numerical methods to find the value of log₂(288).
Assuming that log₂(288) ≈ 8.1699 (rounded to four decimal places), we can rearrange the equation:
5 + n = 8.1699
n = 8.1699 - 5
n ≈ 3.1699
Therefore, the value of n is approximately 3.1699.
Now, let's find the value of m. We use the fact that:
32/3 + m = 0 (since the x-coordinate on the new curve is 0)
m = -32/3
Therefore, the value of m is -32/3.
Finally, we can find m + n:
m + n = -32/3 + 3.1699 ≈ 0.8366
So, the value of m + n is approximately 0.8366.