Every point (x,y) on the curve y=log(3x)/log2 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(12x−96)/log2. What is the value of m+n?

Details
that is log base 2 which i expressed in base 10

since log(12x-96)

= log(12*(x-8))
= log12 + log(x-8)
the translation is (8,log12)
using logs base 2

can you please explain what this question means

Why are you asking Brilliant questions here. The answers will be posted here next week.

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To find the value of m+n, we need to analyze the transformation of the curve and determine the relationship between the original and the transformed points.

Let's start by analyzing the original curve, which is given by the equation y = log(3x)/log2.

The given equation represents a logarithmic function with base 2. Therefore, we can rewrite it in terms of the natural logarithm (base e) to simplify the calculations:

y = (log(3x)/log2) * log(e)/log(e)
= log(3x)/log(e*2)
= log(3x)/ln(2)

Now, we need to find the equation of the transformed curve. The transformation is defined as (x′,y′) = (x-m, y-n), where m and n are integers.

Substituting the values into the transformation equation, we get:
x′ = x - m
y′ = y - n

We are given the equation of the transformed curve, which is y' = log(12x-96)/log2. Replacing the variables with the transformation equations, we have:
log(3(x-m))/ln(2) = log(12(x-m)-96)/ln(2)

Since the bases of the logarithms are the same (ln(2)), we can cancel them out:
log(3(x-m)) = log(12(x-m)-96)

Now, we can equate the expressions inside the logarithms:
3(x-m) = 12(x-m) - 96

Simplifying and rearranging the equation:
3x - 3m = 12x - 12m - 96
-9x + 9m = -96

Dividing both sides by 9:
-x + m = -96/9
-x + m = -32

From this equation, we can see that m = x - 32.

To find the value of m+n, we need to substitute m = x - 32 back into the transformation equation:
m + n = (x - 32) + n

Since m and n are integers, it means that they add up to a constant value regardless of the value of x. Therefore, m + n = constant.

However, since the problem does not provide specific values for x, we cannot determine the exact value of m + n. We can only conclude that m + n is a constant.