User:Guy vandegrift/sandbox

I find this change of base proof easier to remember
https://en.wikiversity.org/w/index.php?title=Speak_Math_Now!/Week_9:_Six_rules_of_Exponents/Logarithms&oldid=2362681#Change_of_base

Derivations are easier to remember if they do not require identities that may see obscure to some readers. Above all, each step must seem obvious. The following derivation relies on definitions that emerge naturally if we use the exponential, $y=b^n$, to define the exponent, $n$, as the logarithm of $y$ in the expression, $y=b^n$. The advantage of this emphasis on exponents is that students can be easily convinced that $b^{n+m}=b^nb^m$ and $b^{kn}=\left(b^n\right)^k$. Using these two facts, we proceed as follows:

From $b^n=B^N$, we randomly select one of the two bases and solve:
 * $b=B^{N/n}$

Now take the logarithm of both sides. It makes no difference whether the base of this logarithm is $B$ or $b$. For example if we take $log_B$, we obtain:
 * $\log_B(b)=\log_B(B^{N/n})=\frac N n \log_B (B)=\frac N n = \frac{\log_B(y)}{\log_b(y)}$ $\Rightarrow$ $\log_b(y)=\frac{\log_B(y)}{\log_B(b)}$

Interchanging the variables $b\leftrightarrow B$ recovers the result that would have been obtained if the other base $(b)$ had been selected.


 * $\log_B(y)=\frac{\log_b(y)}{\log_b(B)}$

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