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Find an expression for log x as a quadratic polynomial in (x-1)

Find constants a,b, and c such that

    \[ \log x = a + b(x-1) + c(x-1)^2 + o((x-1)^2) \qquad \text{as} \qquad x \to 1. \]


From the Taylor formula for \log (1+x) we have

    \[ \log(1+x) = x - \frac{x^2}{2} + o(x^2) \quad \text{as} \quad x \to 0. \]

Replacing x by x-1 we then have

    \begin{align*}  \log (1+x-1) = \log x &= (x-1) - \frac{(x-1)^2}{2} + o((x-1)^2) & \text{as } (x-1) \to 0 \\[9pt]  &= (x-1) - \frac{1}{2}(x-1)^2 + o((x-1)^2) & \text{as } x \to 1.  \end{align*}

Therefore,

    \[ a = 0, \qquad b = 1, \qquad c = -\frac{1}{2}. \]

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