Statistics: Exponential function and power function?

Question

for my math project, im trying to predict the future world record time of the men's 200m and the women's triple jump.

I'm using the equations y=ab^x and y=ax^b. My teacher said each equation is fit to predict the future times of a jumping or a running event.

So what is the difference between the exponential function and the power function?

Answer 1

ln(ab^x) = ln(a) + xln(b)

ln(ax^b) = ln(a) + bln(x)

Both logarithmic functions are straight lines, so a simple variable substitution makes an easy application of the method of least squares. The major difference between the two is that ab^x has a y intercept and ax^b does not.

Answer 2

OK, so we have w=e^v Here, w and v are just numbers. e is Euler's number (I think that's what its called) which is equal to 2.7181... So, say v=2. This means that w=2.7181^2=7.3891 Now, inverting it to solve for v, we get v=log_E (w) This means that v is equal to log base e of w. Basically, if we have a number, w, we can find out what number, v, is needed for e to be raised to the power of to make w. Since I'm a bit tired, I'll do a numerical example, since that probably didn't make sense We let log_E=ln, which stands for natural log, just for simplicity; it is easier to write ln than log_E every time So, example: We have this: e^v=665.1416 So, putting in numbers, we have: 2.7181^v=665.1416 Using the ln function, we get v=ln(665.1416) And using a calculator gives v=6.5. That's it. Basically, all the ln function does it allow you to find what power, v, is needed to get e^v=w :)