Been studying for my Math 131 final. It's kind of hellish, especially since our professor is insisting on teaching us new material right up until we take the final exam, which we take the very last day of the semester. We're currently working on inequalities. But, that's why I've been quiet. 

More of a post will follow but in the mean time, look at the below word problem, and see if you can figure it out. I'll be posting the answer after my final. 

"I have 400 grams of Kryptonite. Kryptonite decays continuously at a rate of 4.5%. Fortunately, it only takes 50 grams to kill Superman. But I need a lot of time to plan my fiendish plot. How much time do I have before the Kryptonite reaches the minimum dose? 

Hint: The mathematic constant e measures continuous growth or decay at exponential rates. So, you want a formula that looks like this: where K is the amount of Kryptonite, t is the time, and r is the rate of decay.

My last exam was on functions. I was pretty happy with it, except for a problem which read like this: 

Easy enough on paper - an function composed with its inverse leaves your plugged in value behind. In this case, the function of 0 inverted is just 0. BUT. f(x) doesn't have an inverse function, cause it's a quadratic. I'm hoping it wasn't a trick question by the professor, and more hoping that we'd recognize that functions are undone by their inverses. The rest of the exam I felt reasonably alright about.

I feel a lot happier today about logarithms than I did a few days ago. I could see they were blue hippos, but I couldn't see whether I could multiply two blue hippos to get a green one. One of my main sticking points, though, is one of the main points of logarithms and their operations - properties of logarithms. Much like properties of exponents! Well, that's cause they are exponents. 

Exponential functions and logarithmic functions are going to be my next exam. Two more exams, then a final. And then, assuming I pass this section of algebra, I'll move on to Math 7A and 7B, pre-calc! Fun times, huh. So. For the people here for math. 

This is an exponential equation. . You can say that as "two to the power of two equals four."
And this is the logarithmic equivalent of that same equation. . That's pronounced "the log of 4 base 2 equals 2." In other words, the number that two gets raised to in order to get four is equal to two. Logarithms are exponents. The argument (the bit in parentheses) is what you end up with when you raise the base (the big subscripted next to the log) to a power (the logarithm itself.) So the logarithm of 27 when the base is 3 is 3, because three to the third power equals 27. 

More...

We've been working with exponential functions. When given the function , where a is some real number, we can draw a nice exponential curve on a graph. For example, the graph of  would have ordered pair coordinates such as

So we end up with a horizontal asymptote at the line x=0 and the following domain and ranges:

Now, let's say we want the inverse of that.  When you want the inverse of a function, the domains and ranges switch. So you end up with:

And to find the inverse of a function mathematically, you replace the f(x) with y, switch the places of the x and y variables, and solve for the y. Simple, right? Let's look at that.

So just solve for y, right? But how do we rewrite that? As my math professor told me today, and I'm still trying to wrap my head around, you cannot write that with algebraic operations. So mathematicians created a new operation. And my math professor demonstrated that new operation with the representation of a blue hippo.

And that is the inverse of our f(x) above. Less likely to eat me than an actual blue hippo, though.