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# Logarithms and Exponential Functions

# Change of Base

Sometimes you'll want to solve a logarithm that isn't base-10 or a natural log. "That's crazy! Madness!" you might scream. We can empathize. If you want to get Log's help in the fight with Expo, though, there's going to be some long, strenuous training sessions ahead. Cue training montage music.

Thankfully, the tool you need to solve these bizarro-logs is pretty easy to use. Like ye olde alchemists of medieval times (not Medieval Times the restaurant, the actual time period), you can actually change the base of the logarithm. Not quite as impressive as changing lead into gold, but we do what we can.

Let's say we've got this nasty looking lump of log:

log_{7} 100

After adding a few powdered pearls, some fire salts, maybe a little old cheese, and some heat we can use this formula:

PLEH! The pearl powder got all over us. We guess we'd better turn the heat down. We can see that this formula allows us to change a log of any base into a fraction of two logs that have a new base of the same value. This new base can be anything we want, as long as it is a proper logarithm base (that is, it's not zero or negative). We'll go easy on ourselves and use base 10.

Let's get on to turning that log to gold:

log_{7} 100

Maybe 2.367 isn't exactly gold, but it'll have to do for now. At the very least, you're one step closer to being able to handle exponential functions.