Logarithms, more!!! MORE!!!!

So far, we’ve learned some pretty properties of logarithms. But nearly all of our logarithm properties/theorems have involved what’s going on inside the logarithm. In other words, they’ve been about the argument rather than the base¹. For example:

• We have a theorem about \(\log_a(b\cdot c)\)
• We have a theorem about \(\log_a\left(b^c\right)\)
• We have a theorem about \(\log_a\left(\frac{b}{c}\right)\)

But all of those things tell us how to deal with the argument of the logarithm. What about the base??? What if we have a logarithm whose base is complicated—are there rules we can come up with to help simplify it? For example, what if we have a logarithm whose base is two things multiplied together—can we simplify that?!? \[\log_{a\cdot b} (c) = \, ???\] What if we have a logarithm whose base is something raised to an exponent—can we do anything with that?!?! \[\log_{a^b}(c) = \, ???\] What if we’ve got two things added together in the base? Can we simplify it? \[\log_{a+b}(c) = \,???\] What about reciprocals?? \[\log_{1/a}(b) = \,???\] And so forth! In other words: \[\boxed{\log_{\left(\substack{\text{but what}\\ \text{about THIS?!?}}\right)}\left(\substack{\text{all of our log theorems}\\ \text{are about THIS}}\right)}\] Try experimenting and playing around to come up with hypotheses for how some of these things might work. For example, we know \(\log_9(81)=2\). How does this relate to \(\log_{3^2}(81)\) and \(\log_{3}(81)\)? What about, say, \(\log_8(4096)\) versus \(\log_{2^3}(4096)\) versus \(\log_2(4096)\)? How about \(\log_{1/3}(9)\) versus \(\log_3(9)\)?? What would be good examples to test to investigate \(\log_{a\cdot b}(c)\)? Can you come up with some more examples to test? Can you come up with some possible theorems? Can you prove those theorems??

And, if you manage any of that, can you solve this cool problem Coltrane showed me? \[\text{solve for }x: \quad \log_5(x)+\log_{\sqrt{5}}(x) + \log_{\frac15}(x) = 6\]