Log Identity Proofs: Simplifying a^{log_{b}(c)}=c^{log_{b}(a)}

[K29]

Homework Statement

a^{log_{b}(c)}=c^{log_{b}(a)}

The Attempt at a Solution

Take log_{a} of both sides:
log_{a}(a^{log_{b}(c)})=log_{a}(c^{log_{b}(a)})

gives:
log_{b}c=log_{b}alog_{a}c

Looks like one more step for the RHS. I sort of see that the RHS should become log_{b}c and then we'll be done. But standard log laws don't seem to help me make this step. Please help with this step/explain why this is so?

 

[SammyS]

Homework Statement

a^{log_{b}(c)}=c^{log_{b}(a)}

The Attempt at a Solution

Take log_{a} of both sides:
log_{a}(a^{log_{b}(c)})=log_{a}(c^{log_{b}(a)})

gives:
log_{b}c=log_{b}alog_{a}c

Looks like one more step for the RHS. I sort of see that the RHS should become log_{b}c and then we'll be done. But standard log laws don't seem to help me make this step. Please help with this step/explain why this is so?

Using the commutative law of multiplication, the RHS is \displaystyle \ (\log_{a\,}c)\,(\log_{b\,}a)\quad\to\quad\log_{b\,}\left(a^{\log_{a\,}c}\right)

 

[K29]

Got it. Thanks a bunch