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Algebra Difficulty 2.7 Junior Find the answer

Given logap=logbq=logcr=logx\frac{\log{a}}{p}=\frac{\log{b}}{q}=\frac{\log{c}}{r}=\log{x}, all logarithms to the same base and x1x \not= 1. If b2ac=xy\frac{b^2}{ac}=x^y, then yy is:

Pick one

Solution

We are given:
b2ac=xy\frac{b^2}{ac} = x^y
Taking the logarithm on both sides:
log(b2ac)=logxy\log{\left(\frac{b^2}{ac}\right)} = \log{x^y}
Using the properties of logarithms:
2logblogalogc=ylogx2\log{b} - \log{a} - \log{c} = y \log{x}
Substituting the values given in the problem statement:
2qlogxplogxrlogx=ylogx2q \log{x} - p \log{x} - r \log{x} = y \log{x}
Since x1x \neq 1, dividing each side by logx\log{x} we get:
y=(C) 2qpry = \boxed{\textbf{(C) } 2q - p - r}
~ proloto

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.