Logarithms and Exponential Functions Introduction
"Look, there he is!" The paparazzi start snapping pictures and the fans scream as the new star steps out of his black stretch limousine. News reporters are in attendance to broadcast the event: "Mr. Expo Nential is finally here at the Mathtastic! Awards, brought to you by Shmoop. Everyone is expecting big things from him. Really big things." Mr. Expo flashes a big smile to the cameras.
"The hopeful ladies in the crowd are going to be disappointed today, though. Looks like Mr. Exponential has his new girlfriend along for the ride, Ms. Log Arithm," the newsman says. "Weird name, I know. By all accounts, Expo and Log are totally different from each other. As they say, though, sometimes opposites attract."
If you'd like an in-depth review of exponents, check out that topic in our guide here. Even if you think you know exponents inside-and-out, a quick refresher never hurt anyone.
Just like multiplication is a shortcut to help with addition (2 + 2 + 2 + 2 = 2 × 4), exponents are shortcuts to help with multiplication (2 × 2 × 2 × 2 = 24). Expo made his name by doing the same things other people did, but making it look really easy.
You've learned all about exponents in earlier chapters, and now we'll get to show you what's what with exponential functions. You might be wondering, "What are they? What are they used for? Why am I here?" Hold onto the velvet rope, and we'll show you the way. All you need to know for now is that in an exponential function, its independent variable, the x, is in—you guessed it—the exponent. See, you've got this down already.
We know all about Expo and his biz, but what about this new girlfriend of his? Log, or logarithm as her mom calls her, helps Expo see his true self. Even he gets a little confused at times and needs some help. Sometimes when evaluating an exponential function, it's hard to solve for x. That's where logarithms swoop in and save the day, like a mathematical superhero. (Logman, maybe?)
Not only can they be used to solve exponential functions, logarithms can also be in functions of their own. Think of it like their secret lair. (Logcave?) You'll learn about all that and more in just a few camera flashes.
So brush off your formal attire, it's time to walk down the red carpet.