The change of base formula

I know the definition of the change of base formula, but I really don't know why it's actually called the "change of base" formula. I know that it allows you to solve for an exponent when the original log is not in terms of base 10 or e. For example: log3(5) = log(5)/log(3). Besides solving for the exponent, what exactly did I do here? Where exactly was a base changed? Was this supposed to change the base of the original log? Sorry if this sounds stupid. Here is a definition I read: "A formula that allows you to rewrite a logarithm in terms of logs written with another base." So does this mean that I could change: log4(x) in in terms of base 2? I could write log2^2(x), but I don't think that would help much? If I used the change of base formula, it would be: log2(x)/log2(4), which just creates a fraction.