Can you adapt the ideas in the proof to produce convincing proofs of the following results?

\[ \log_c a - \log_c b = \log_c \frac{a}{b}\]

\[ n\log_c a = \log_c a^n\]

\[\dfrac{1}{n}\log_c a = \log_c a^{\frac{1}{n}} \]

\[(\log_a b) \times (\log_b c)=\log_a c\]

What conditions must \(a,b,c,n\) satisfy?