Here’s another description:
Say we have 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, means that . If we want this in base c, that means we want where . Now check this out…we know that …so we can just say, lets make where . Then that would mean that since and . Now, we know how to solve an exponent of an exponent:.
So, we want , where we know and and thus . So, and , thus . Then since , , and / .
Finally! We’ve converted to another base, the base of !
It may have been easier to label the starting logarithm as . Then . And . This may be easier to visualize: say and . That means that and . Then . Then by the power rule of logarithms, and since , that means , which means that .
Why does the power rule of logarithms work? Say and . Say g=3. So means which is (a)(a)(a)=b. Then , and say h=2. So 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 . Since h=2 and , that’s 2*3=6, as shown by the result (a)(a)(a)(a)(a)(a).