Exponents & Logarithms

Since Elven picked me to be scribe I SUPPOSE I have to blog for today's class..Thanks Elven and isn't it your birthday tomorrow? LoL But anyways..

We started off class by graphing the function f(x) = 2^(x-2) and also finding the domain, range, horizontal asymptote and both x & y intercepts. On slide 2 it shows that the domain is (-∞,∞), the range is (0,∞), the horizontal asymptote is y=0, the y intercept is 1/4 and there is no x intercept.

On the next slide (3) we had to graph the function f(x) = -2^(x+2) a bit different but the same concept applies. The domain is (-∞, ∞), the range (-∞, 0), the horizontal asymptote is y=0, the y intercept is -4 and there is no x intercept. It also helps if you make a table of values to know where each point is but that is just how I prefer to do it.

On the next slide (4) we had to graph the inverse of f(x) = 2^(x-2). Symbolically, it looks like this x = 2^(y-2). Basically you're looking for the y value. Graphically, you flip the orginal function diagonally where it flips over the line of reflection. On slide 4, it shows the blue dotted line is y = x (line of reflection) where the inversed function is flipped over. The green indicates the inverse function.

On slides 5, 6, and 8 we were to rewrite each table of values for the inverse of each function. Remember that the inputs turn into the outputs and vice versa. The first thing you should do is figure out the table of values of the orginal function then the inverse of each function. For example, on slide 6 it shows that X is -3 and the function is 2^x then the output would be 1/8. Therefore the inverse of it would be, x = 1/8 and f^y(x) = -3 because the output of the inversed function is the input of the exponential function and vice versa.
Another way of looking at it is that in exponential functions the exponets turn into powers but in inverse functions, powers turn into exponents. On slide 7 it shows 2^3 = 8 in which the 2 is the base, the 3 is the exponent and the 8 is the power. This is called the anatomy of a power.

Pretty easy right? However, according to Mr. K, we're all going to forget though he stresses that we shouldn't. Hopefully no one will forget. Cross fingers!

Anyways.. on slide 9 we had to graph the inverse of 2^x. The original function is in black, the inverse function is the green, and in blue is the line of reflection. To find the inverse function you take points from the exponential function such as (0,1), (1,2), and (-1, 1/2) and switch the x and y values to get the points for the inverse function. Therefore the points would be (1,0), (2,1), and (1/2,-1). Then all you have to do is connect the dots. The name for this function is apparently called log2^x when it could've been called something else like George or Peter. Unfortunately it's not..

On slide 10, it describes that a logarithm is an exponent. On the left side, it shows b^a = c in which b is the base, a is the exponent and c is the power. However on the right side it shows, logb^c = a in which b is the base, a is the (exponent) logarithm and c is the power (argument). DO NOT FORGET!!! Even though we will says Mr. K!

Where this applies is on slides 11, 12, and 13. One of the examples are log3^9 = 2 because what you are looking for is the exponet which we already found. But in order to do this you must remember that exponents turn into powers and powers turn into exponents in inverse functions. Basically, you're looking for an exponent that would give you 9 with the base being 3. However there is one special case that if the power is 1 the exponent will always be zero because anything to the power of zero is 1. If you get a question like log16(8), in order to this you let it equal to x which changes it to 16^x = 8. From the previous lessons we were taught to find the same bases and figure out the exponents using algebra. Therefore you get the answer x = 3/4(refer to slide 12).

On the last slide it's a bit different because we are to write each in exponential expressions. The second example is log2(16) = 4 because what you're looking for is what exponent equals 16 with the base of 2 which is 2. Therefore you write it as 2^4 = 16.

WELL.. I believe I am finished, finally! Time to watch some Girlicious, but before I forget the next scribe will be Joyce! Toodles!