[math]
q > 0, \ p > 0, \ p \ne 1, r > 0, r \ne 1, \text { and } y = \log_p(q) \implies \\
p^y = q \text { by the definition of logarithms.}\\
p^y = q \implies\ log_r(p^y) = \log_r(q) \implies y \log_r(p) = \log_r(q).\\
p > 0 \text { and } p \ne 1 \implies \log_r(p) \ne 0.\\
\therefore \ y \log_r(p) = \log_r(q) \implies y = \dfrac{\log_r(q)}{\log_r(p)}.\\
\text {But } y = \log_p(q) \text { by hypothesis.}\\
\therefore \ \log_p(q) =\dfrac{\log_r(q)}{\log_r(p)}.
[/math]
