Exercise 3.2-2

Prove equation (3.16).

Equation 3.16 says: \(a^{\log_b c} = c^{\log_b a}\)

This can be shown in many ways, all of which requires rewriting and manipulating logarithmic identities that were listed in the book just before equation 3.16.

Method 1

\[\begin {aligned} a^{\log_b c} & = (b^{\log_b a})^{\log_b c} \\ & = (b^{\log_b c})^{\log_b a} \\ & = c^{\log_b a} \end {aligned}\]

Method 2

\[\begin {aligned} a^{\log_b c} & = a^{\log_a c/\log_a b} \\ & = \big(a^{\log_a c}\big)^{1 / {\log_a b}} \\ & = c^{1 / {\log_a b}} \\ & = c^{\log_b a} \end {aligned}\]