In the order of operations, the “Exponents” rule represents a bunch more than just superscripts or tiny numbers flying up and to the right of things.

Roots are exponents, too!

Not the ones from trees, but things like square roots and cube roots. Consider render1. You do the square root first because it qualified as an “exponent.”

But if you had , the 9 + 2 is under the radical sign (the square root sign) so it’s bound together in the “Parenthesis” rule.

This one isn’t that hard with arithmetic, but when you come to algebra and start “undoing” these things – it’s important to remember that roots fall into this category.

Fractional exponents are exponents.

This one seems pretty “duh” so it’s easy to see how they fall into the “E” of the order of operations. But what are fractional exponents really?

means .

So fractional exponents are the same as roots.

Note that some fractional exponents are roots and “plain” exponents all mixed up. Like this one:

means as well as .

This is a big fat full concept that needs a little more explaining. So I’ll write more on these in another article.

Logs fall under the E.

As my algebra and computer math teacher in high school, Mrs. Kelley, used to tell us – logarithms are exponents. It took me a long time to figure out what the heck she meant. But when I did, I thought it was brilliant.

Notice who the exponent is in this: 3^{2} = 9. 2 is the exponent. And 2 is the same as because the equals sign in means “is the same as.” So the logarithm is the exponent 2.

Still with me? Either way, it’s okay. It’s a weird concept that I can go into detail in a video soon.

The thing to remember here is that logarithms fall into the “Exponents” rule of the order of operations.

So if you have you have to do the first and then add the 7 after.

Want more on exponents?

In the meantime, you can check out more than everything you always wanted to know about exponents on the Wikipedia Exponents page. Rebecca Zook created a great video on logarithms. And check out this explanation and problems to work on fractional exponents.

And let me know what you think. Did I miss something?

Thanks so much for linking to me and for the shoutout! Kate Nowak also had a great post up about how to introduce logs without freaking students out, if you’re intereseted: http://function-of-time.blogspot.com/2009/12/introducing-logs.html

It’s a great approach and I’ve used it successfully with my own tutoring students!

Thanks for the link, Rebecca! Kate’s suggestion is awesome. I’ll start using it. I even tweeted it!

Another thing I sometimes do is teach the International Log Song – totally unrelated to math logs, but fun to sing. Totally loosens the students up! http://www.youtube.com/watch?v=zkNj1WXqDmw&feature=related