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Question Number 193767 by sciencestudentW last updated on 19/Jun/23

$$provethat{c}^{{log}_{b}a}=a^{{log}_{b}c}$$

Answered by deleteduser1 last updated on 19/Jun/23

$$c^{{log}_{b}a}=c^{\frac{loga}{logb}=}{\left(c^{\frac{{log}_{c}a}{{log}_{c}10}}\right)}^{\frac{1}{logb}}={\left(a^{\frac{{log}_{c}c}{{log}_{c}10}}\right)}^{\frac{1}{logb}}={\left(a^{logc}\right)}^{\frac{1}{logb}}$$$$=a^{\frac{logc}{logb}}=a^{{log}_{b}c}$$

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