If a, b, c are positive then the minimum value of alog⁡b−log⁡$$c+blog$$⁡c−log⁡$$a+clog$$⁡a−log⁡b is

Question

If a, b, c are positive then the minimum value of  alog⁡b−log⁡$$c+blog$$⁡c−log⁡$$a+clog$$⁡a−log⁡b is

Infinity Learn · Accepted Answer

Since A.M. ≥ G.M.∴ alog⁡b−log⁡$$c+blog$$⁡c−log⁡$$a+clog$$⁡a−log⁡$$b3$$≥alog⁡b−log⁡c⋅blog⁡c−log⁡a⋅clog⁡a−log⁡$$b3$$           (1) Let  $$x=alog$$⁡b−log⁡c⋅blog⁡c−log⁡a⋅clog⁡a−log⁡b⇒log⁡$$x=(log$$⁡b−log⁡$$c)log$$⁡$$a+(log$$⁡c−log⁡$$a)log$$⁡$$b+(log$$⁡a−log⁡$$b)log$$⁡c⇒  $$logx=0$$  ⇒$$x=1$$ ∴ From (1), alog⁡b−log⁡$$c+blog$$⁡c−log⁡$$a+clog$$⁡a−log⁡b≥$$3$$.

If a, b, c are positive then the minimum value of $a^{\log b - \log c} + b^{\log c - \log a} + c^{\log a - \log b}$ is

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If a, b, c are positive then the minimum value of alog⁡b−log⁡c+blog⁡c−log⁡a+clog⁡a−log⁡b is

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If a, b, c are positive then the minimum value of $a^{\log b - \log c} + b^{\log c - \log a} + c^{\log a - \log b}$ is