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Two variables, x and y, are connected by the equation y = ab^x - Scottish Highers Maths - Question 12 - 2019

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Question 12

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Two variables, x and y, are connected by the equation y = ab^x. The graph of log2 y against x is a straight line as shown. Find the values of a and b.

Worked Solution & Example Answer:Two variables, x and y, are connected by the equation y = ab^x - Scottish Highers Maths - Question 12 - 2019

Step 1

State linear equation

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Answer

From the given equation, we can take the logarithm base 2 of both sides. This gives us:

extlog2y=extlog2(abx) ext{log}_2 y = ext{log}_2(ab^x)

Using the laws of logarithms, this can be expanded to:

extlog2y=extlog2a+ximesextlog2b ext{log}_2 y = ext{log}_2 a + x imes ext{log}_2 b

This indicates that log2 y plotted against x is a straight line, where:

  • The intercept is extlog2a ext{log}_2 a
  • The gradient is extlog2b ext{log}_2 b

Step 2

Interpret intercept

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Answer

From the graph, we note that when x = 0, log2 y = -1. Therefore, substituting into our equation:

ext{log}_2 a = -1 \\ a = 2^{-1} = rac{1}{2}

Step 3

Interpret gradient

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Answer

The gradient from the graph is also observed to be 3. Thus, we can state:

extlog2b=3b=23=8 ext{log}_2 b = 3 \\ b = 2^3 = 8

Step 4

State a and b

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Answer

Finally, we can conclude that:

  • The value of a is rac{1}{2}.
  • The value of b is 8.

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