Why the Change of Base Formula Works For Logarithms and Exponents

I find I remember math concepts so much more if I understand why and how things actually work.  That said, why does the change of base formula work?  Check out the proof in this Wikipedia article.

Here’s another description:

Say we have y=log_a b and you want to change the base to c, for example, say you only have a log base c button on your calculator, such as log base 10.  Well, y=log_a b means that a^y= b.  If we want this in base c, that means we want c^z= b where z=log_ c b.  Now check this out…we know that a^y= b…so we can just say, lets make c^v= a where v=log_c a.  Then that would mean that (c^v)^y=b since c^v= a and a^y= b.  Now, we know how to solve an exponent of an exponent:(c^v)^y=c^{vy}.

So, we want c^{vy}, where we know v=log_c a and y=log_a b and thus vy= log_c a * log_a b.  So, c^z= b and c^{vy}=b, thus log_c b=vy.  Then since vy= log_c a * log_a b, log_c b=log_c a * log_a b, and log_c b/log_c a= log_a b.

Finally! We’ve converted y=log_a b to another base, the base of c!

It may have been easier to label the starting logarithm as y=log_a x. Then y=log_a x=log_a b * log_b x. And log_b x=log_a x/log_a b. This may be easier to visualize: say g=log_a b and h=log_b x. That means that a^g=b and b^h=x. Then x=b^h=(a^g)^h. Then y=log_a x=log_a (a^g)^h= h * log_a (a^g) by the power rule of logarithms, and since log_a (a^g)=g, that means y=log_a x=g * h, which means that y=log_a x=g * h=log_a b * log_b x.

Why does the power rule of logarithms work? Say a^g=b and b^h=x. Say g=3. So a^g means a^3 which is (a)(a)(a)=b. Then b^h=x, and say h=2. So b^h=b^2 means (b)(b). Well, since (a)(a)(a)=b, (b)(b)=(a)(a)(a)(a)(a)(a), see the associative property of multiplication for more info. So, that demonstrates an example of how y=log_a x=log_a b^h=h * log_a b. Since h=2 and log_a b=g=3, that’s 2*3=6, as shown by the result (a)(a)(a)(a)(a)(a).

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2 Responses

  1. Thanks for that. Now I understand.

  2. Very good post.Really looking forward to read more. Great.

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