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Find all real solutions to the equation: log_(0.25)((x² + 2x - 8)²) - log_(0.5)(10 + 3x - x²) = 1.

1. (sqrt(331) - 1) / 6
2. (-sqrt(331) - 1) / 6
3. (sqrt(73) - 7) / 2
4. (-sqrt(73) - 7) / 2

Correct answer: (sqrt(73) - 7) / 2

Solution

Using base-change: log_(0.25)(A²) = log_((1/2)²)(A²) = (2/2)*log_(1/2)(A) = log_(1/2)(A). Wait: log_(0.25)(y) = log(y)/log(0.25) = log(y)/(-2*log2). And log_(0.5)(y) = log(y)/log(0.5) = log(y)/(-log2). So log_(0.25)(y) = (1/2)*log_(0.5)(y). Then the equation becomes: (1/2)*log_(0.5)((x²+2x-8)²) - log_(0.5)(10+3x-x²) = 1, i.e. log_(0.5)(|x²+2x-8|) - log_(0.5)(10+3x-x²) = 1.

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