Study Guide

Logarithms and Exponential Functions

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. Expo Nential 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 (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.

Logarithms and Exponential Functions Resources


Logarithms: Introduction to "The Relationship"
They make it sound like logarithms and exponents are going steady. (In a very serious way.) This page gives an excellent overview of how logarithms and exponents are related, and how they can be simplified.

Inverse Functions
Feel like getting your world turned upside-down again? We sure do. Strap yourself in and bring a doggy bag if necessary, because this site focuses all on inverse functions and operations.

Regents Prep: Exponential Functions
If graphs really get you fired up, this site is for you. It contains a solid re-visit of exponential functions and how they are graphically different from logs.


TED Ed Lessons: Logarithms Explained
A video with a breakdown of logarithms using wiggly, wobbly numbers and a set of really red eyes. It's also got a good example of how logarithms apply to chemistry.

AOL On: How to Simplify Logarithms
If you'd like to bone up on some of the basic rules of simplifying logarithms with a helpful narrator to guide you through, this is the place.

Exponential Growth Functions
This video has a good overview of graphic exponential functions, and a demonstration of how exponential functions never go to the dark side. Er, negative side.

Online Math Learning: Exponential and Logarithmic Functions
We heard there was a bulk sale on math videos over here. Practically anything you can imagine involved with exponentials and logs is explained in moving-picture form for your enjoyment.


Quia: Rags to Riches Exponents and Logs Game
Who wants to be a fake millionaire? We knew you would. This game is a quiz-show style affair that tests your knowledge of exponential and logarithmic functions. Three lifelines, no phone-a-friend—can you win?

Lake Tahoe Community College: Exponential Equations
Here's an interesting take on solving exponential equations. It's like a choose-your-own-adventure book for efficient solving methods. If you want to sharpen your skills at recognizing the best path to take, this is the place to do it.