A1

Good to know

1. Logarithm Exponent Swap Rule

The rule states that you can swap the base of a power with the argument of its logarithmic exponent.

$a^{\log_b c} = c^{\log_b a}$

This is why $12^{\log n}$ is the same as $n^{\log 12}$.

Proof Outline

The proof works by taking the log of both sides and showing they are identical.

1. Start with one side of the identity: $a^{\log c}$
2. Take the log of the entire expression: $\log(a^{\log c})$
3. Apply the Power Rule for logs ($\log(x^y) = y \log x$): $(\log c) \cdot (\log a)$
4. Use the Commutative Property of multiplication to swap the terms: $(\log a) \cdot (\log c)$
5. Apply the Power Rule in reverse: $\log(c^{\log a})$
6. Conclusion: Since $\log(a^{\log c}) = \log(c^{\log a})$, the original expressions must be equal.

Key Idea: The proof hinges on the fact that multiplication is commutative ($\log a \cdot \log c = \log c \cdot \log a$).

2. Proof by Induction

[https://www.mathcentre.ac.uk/resources/uploaded/mathcentre-proof.pdf]