Prove that $b^{\log_d(a)}=a^{\log_d(b)}$ and $\log_a(b)*\log_b(c)=\log_a(c)$

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How to prove $b^{\log_d(a)}=a^{\log_d(b)}$ and $\log_a(b)*\log_b(c)=\log_a(c)$

I've found this properties but the book doesn't include the proofs of them. Can you help me please?