Given that
$$\log_2 x^3 - \log_2 y^2 = 9$$
show that
$$x = A y^p$$
where A is an integer and p is a rational number. - AQA - A-Level Maths Pure - Question 9 - 2022 - Paper 2
Question 9
Given that
$$\log_2 x^3 - \log_2 y^2 = 9$$
show that
$$x = A y^p$$
where A is an integer and p is a rational number.
Worked Solution & Example Answer:Given that
$$\log_2 x^3 - \log_2 y^2 = 9$$
show that
$$x = A y^p$$
where A is an integer and p is a rational number. - AQA - A-Level Maths Pure - Question 9 - 2022 - Paper 2
Step 1
Using the properties of logarithms
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Answer
Applying the logarithmic property that states (\log_b M - \log_b N = \log_b \left( \frac{M}{N} \right)), we can rewrite the initial equation:
log2(y2x3)=9.
Step 2
Exponentiating both sides
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Answer
To remove the logarithm, we exponentiate both sides of the equation. This gives us:
y2x3=29
which simplifies to:
y2x3=512.
Step 3
Rearranging the equation
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Answer
Multiplying both sides by (y^2) leads to:
x3=512y2.
This can be rearranged to:
x3=83y2.
Step 4
Expressing x in the desired form
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Answer
Taking the cube root of both sides results in:
x=8⋅y32.
Here, we identify (A = 8) (an integer) and (p = \frac{2}{3}) (a rational number), satisfying the requirement of the question.
