Good Day, We know these facts: $\displaystyle x = log_a(bc)$ $\displaystyle y = log_b(ca)$ $\displaystyle z = log_c(ab)$ The equation below must be proved: $\displaystyle x + y + z + 2 = xyz$
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Consider $\displaystyle bc = a^x ~,~ ca = b^y ~,~ ab=c^z $ we have $\displaystyle a^{xyz} = (bc)^{yz} = (b^y)^z (c^z)^y = (ca)^z (ab)^y = a^{z+y} b^y c^z = a^{z+y} (ca)(ab) = a^{2+z+y} bc = a^{x+y+z+2} $ Thus , $\displaystyle xyz = x+y+z+2 $
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