Question

Please factorize
`\ln ^2\left(4-x\right)+\ln \left(4-x\right)\ln \left(x+(1)/(2)\right)-2\ln ^2\left(x+(1)/(2)\right)=0`

Answer 1

• [ln (4 - x) - ln (x + 1/2)] × [ln (4 - x) + 2 ln (x + 1/2)] = 0

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Given equation:

• ln² (4 - x) + ln (4 - x) ln (x + 1/2) - 2 ln² (x + 1/2) = 0

Substitute:

• ln (4 - x) = p, ln (x + 1/2) = q

Then we have:

• p² + pq - 2q² =
• p² - q² + pq - q² =
• (p + q)(p - q) + q(p - q) =
• (p - q)(p + q + q) = 0
• (p - q)(p + 2q) = 0

Substitute back:

• ln² (4 - x) + ln (4 - x) ln (x + 1/2) - 2 ln² (x + 1/2) = 0
• [ln (4 - x) - ln (x + 1/2)] × [ln (4 - x) + 2 ln (x + 1/2)] = 0