We looked at logarithms that had fractional and negative answers today. To get a log of a fraction, we need a negative answer. If the base of a log is larger than the number we are taking the log of, then we need the answer to be a fraction. This is because a fractional exponent tells us to take the root of a number. We looked at a few examples of these, and Alexa started to understand the process behind it.
A logarithmic function is the inverse of an exponential function. We can write functions exponentially as b^x=a, and then logarithmically as log base b of a = x. I familiarized Alexa with how to go back and forth between the two forms. She took notes from the board, and we went through several examples. We also started to calculate logarithms. She understood the concepts and thinks that what we are doing so far is easy.
Alexa and I wrapped up her studies of graphing exponential growth and decay today. As we were doing the last of the problems, she asked for more of them, and said that she decided she likes them! Yippee! She was able to graph them by plotting the y-intercept (where x=0), and the point where x = 1 and she found the corresponding y value.
We began graphing exponential decay functions of the form y=ab^x, where a>0 and 0<b<1. In other words, b is a fraction. We graphed them the same way we graphed exponential growth, by letting x=0 and x=1. When x=0, y=a, so the y intercept is (0,a). Then you compute the y value for x=1. Alexa was able to identify if functions were growth or decay by looking at their equations and examining the b value. She completed exercises 3-8 on p. 489.
We extended our study of exponential functions by looking more closely at growth functions of the form f(x) = ab^x. Through our examples, I taught Alexa a shortcut to graphing. Rather than doing a chart with x values from -2 to 2, we can simply use x=0 and x=1. We find that when x=0, the y-intercept for f(x)=b^x is simply (0,1). For the graph of f(x)=ab^x, when x=0, the y=intercept is (0,a). We can then algebraically find the values when x=1, and plot the growth or decay function with just those 2 points. Alexa completed notes and p.482 (7,9,10,12,13).
