Extension: Making Sense of Irrational Exponents

Difficulty Going Beyond Integer and Fractional Exponents

So now that we've seen how to do integer exponents and fractional exponents, the question comes up: How do we do exponents to other powers like irrational numbers?

What if we wanted to take $2^{\pi}$?

Does this even make sense? Think about it for a bit, the answer isn't necessarily obvious.

The answer is we can make it make sense. However we have to think about exponents differently. Usually teachers will tell you that $2^x$ is $x$ copies of $2$ all multiplied together. Which makes sense for $x=3$. We take $3$ twos, and get $2^3= 2\cdot 2\cdot 2=8$. But it doesn't make so much sense for $2^{\pi}$.

So we know we are going to have to think about exponents differently. Let's try thinking about properties of exponents.

These are the two most important properties of exponents. So now if we can find a function that is equal to $2^x$ when x is an integer, and satisfies these properties, then we have found a way to compute exponents for weird decimal numbers as well.

It turns out there is such a function and it looks like this:

So how did we get this function?

That's kind of complicated. It requires a bit of calculus to understand properly.

Intuitively, all exponential functions are fundamentally "the same" in some sense. The most basic one is $e^x$, where $e=2.718281828...$ is Euler's number. All other exponential functions can be found by multiplying the x by a constant in the exponent. To find this constant we find the number that when input into the function gives our desired base. We call this the logarithm, denoted $\ln$. For example $2^x = e^{(x\cdot\ln 2)}$, where $\ln 2$ is the number such that $e^{\ln 2}=2$. This actually comes from one of our properties of exponents above which says that $2^x=(e^{\ln 2})^x=e^{(x\cdot\ln 2)}$.

Blurb

What is the true meaning of a general real exponential $B^x$, for $B>0$ a positive base and $x$ an arbitrary exponent, possibly irrational? This is a good question. One place to start reading is the other explanations here, or an old Math Stack Exchange question.