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

prove that c^(log_b a) =a^(log_b c)

$${prove}\:{that}\:{c}^{{log}_{{b}} {a}} ={a}^{{log}_{{b}} {c}} \\ $$

Answered by deleteduser1 last updated on 19/Jun/23

c^(log_b a) =c^(((log a)/(log b))=) (c^((log_c a)/(log_c 10)) )^(1/(log b)) =(a^((log_c c)/(log_c 10)) )^(1/(log b)) =(a^(log c) )^(1/(log b))   =a^((log c)/(log b)) =a^(log_b c)

$${c}^{{log}_{{b}} {a}} ={c}^{\frac{{log}\:{a}}{{log}\:{b}}=} \left({c}^{\frac{{log}_{{c}} {a}}{{log}_{{c}} \mathrm{10}}} \right)^{\frac{\mathrm{1}}{{log}\:{b}}} =\left({a}^{\frac{{log}_{{c}} {c}}{{log}_{{c}} \mathrm{10}}} \right)^{\frac{\mathrm{1}}{{log}\:{b}}} =\left({a}^{{log}\:{c}} \right)^{\frac{\mathrm{1}}{{log}\:{b}}} \\ $$$$={a}^{\frac{{log}\:{c}}{{log}\:{b}}} ={a}^{{log}_{{b}} {c}} \\ $$

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