Logarithms

Algebra Level pending

If x x and y y are positive numbers such that x 2 + y 2 = 6 x y x^2 + y^2 = 6xy , then which of the following equations must be true?

2 log ⁡ ( x + y ) = log ⁡ x + log ⁡ y + 3 log ⁡ 2 2 \log( x + y) = \log x+ \log y + 3 \log 2 2 log ⁡ ( x + y ) = 3 log ⁡ 3 + log ⁡ x + log ⁡ y 2 \log( x + y) = 3 \log 3 + \log x + \log y log ⁡ x / y = log ⁡ x − log ⁡ y \log x / y = \log x - \log y log ⁡ x y = log ⁡ x + log ⁡ y \log xy = \log x + \log y

Bhuvana Chaithanya by Bhuvana Chaithanya , 19, India

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1 solution

given x^2 + y^2 = 6xy = x^2 + y^2 +2xy = 6 xy+2 xy = (x + y)^2 = 8 xy taking logarithms on both sides =2 log (x + y) = log 8+log x+ log y [log x^m = m log x] [log xy = log x + log y] = log 2^3 + log x +log y =2 log (x + y) = log x + log y + 3 log 2

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It is not clear to me what you are asking for.

Note that x = y = 0 x = y = 0 is a solution to the equation in the problem, but no in the answer options.

Calvin Lin Staff - 4 years, 6 months ago

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I guess the maker has overlooked the fact that x = y = 0 is a solution to the equation, because when using that logarithmic approach it doesn't give that you that solution since log (0) is invalid.

Peter van der Linden - 4 years, 6 months ago

This problem seems very poorly worded...

Tina Sobo - 4 years, 6 months ago

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