Problem 27 · 2024 Math Kangaroo
Stretch
Algebra & Patterns
substitution
It is known that the statements \(2^x=3\), \(2^y=7\) and \(6^z=7\) are true. Which of the following relationships is therefore correct?
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Answer: A — \(z=\dfrac{y}{1+x}\)
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Hint 1 of 2
Rewrite 6 as 2·3 so that 6ᶻ can be expressed using the known powers of 2.
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Hint 2 of 2
Take logs base 2: x = log₂ 3, y = log₂ 7, and 6ᶻ = 7 gives z(1 + x) = y.
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Approach: take logarithms base 2
- From 2ˣ = 3 and 2ʸ = 7, x = log₂ 3 and y = log₂ 7.
- Since 6ᶻ = 7 and 6 = 2·3, we get z(log₂ 2 + log₂ 3) = log₂ 7, i.e. z(1 + x) = y.
- So z = y/(1 + x).
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