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Question Number 208263 by efronzo1 last updated on 09/Jun/24

Answered by Ghisom last updated on 09/Jun/24

$$a={\mathrm{e}}^{\alpha }\Rightarrow b={\mathrm{e}}^{\lambda }a\wedge c={\mathrm{e}}^{2\lambda }a$$$$A={\log }_{c}a=\frac{\alpha }{\alpha +2\lambda }$$$$B={\log }_{b}c=\frac{\alpha +2\lambda }{\alpha +\lambda }$$$$C={\log }_{a}b=\frac{\alpha +\lambda }{\alpha }$$$$B-A=C-B=\delta \Rightarrow$$$$\frac{(3{\alpha }^2+3\alpha \lambda -2{\lambda }^2)\lambda }{\alpha (\alpha +\lambda )(\alpha +2\lambda )}=0$$$$\lambda =0\Rightarrow a=b=c\wedge \delta =1$$$$\lambda =\frac{3\pm \sqrt{33}}{4}\alpha \Rightarrow \delta =\frac{3}{2}$$

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