The equation a^{log_{b}(c)}=c^{log_{b}(a)} can be simplified by taking log_{a} of both sides, resulting in log_{b}c=log_{b}alog_{a}c. The right-hand side can be further manipulated using the commutative law of multiplication, leading to log_{b}(a^{log_{a}c}). This transformation clarifies the relationship between the logarithmic expressions and confirms the equality. The discussion emphasizes the importance of understanding logarithmic properties to simplify complex equations effectively.
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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 said: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?