Presentation on theme: "Technology: A Portal to Exploration and Discovery GCTM October 18th, 2012 Kenn Pendleton"— Presentation transcript:
Technology: A Portal to Exploration and Discovery GCTM October 18th, 2012 Kenn Pendleton email@example.com
Background: Student experiences prior to this exploration Using natural number exponents, the three laws of exponents were developed through exploration. This likely was accomplished using a scientific calculator.
Background: Student experiences prior to this exploration An exponent of zero was explored. A conjecture of the definition of a zero exponent was made and confirmed by applying the second law of exponents. The fact that the laws of exponents still held when exponents of zero were used was verified.
Background: Student experiences prior to this exploration Negative exponents were explored. A conjecture of the definition of a negative exponent was made and confirmed by applying the second law of exponents. The fact that the laws of exponents still held when negative exponents were used was verified.
Arithmetic is done in Run-Matrix. Enter “1,” or cursor to the icon and EXE.
The square root symbol is in gold above the x2 key (SHIFT: x2). Find the square root of 8. To change from simplest radical (F)orm to (D)ecimal form: F↔D (above “8”). Return to the Graph window (MENU; 5).
Thus far all fractional exponents have had a numerator of one. Having seen that the laws of exponents hold with those fractional exponents enables a conjecture about fractional exponents whose numerators are not one. This conjecture can be verified using the graphs already created, and then a general definition of a fractional exponent can be created.
What are logarithms, and how do they relate to the previous exploration? The logarithmic expression “logb n = x” is read “the logarithm, in base b, of the number n is x.” As was true for exponential functions, the base of a logarithm can be any positive number. When a logarithm has a base of 10, the base is not written. Thus, “log x” means the same as “log10 x” and known as the common logarithm.
F1 F2 F3 F4 F5 F6 EXIT to the Graph window; unselect Y1 and Y5. Select Y6, Y7, and Y8.
F1 F2 F3 F4 F5 F6 Change the view window. With no graph, SHIFT ; F3.
X,θ,T There appears to be a reflection across y = x. X,θ,T If so and a point (a, b) were found on y = 10x, the point (b, a) would be found on y = log x.
X,θ,T Discuss what is happening on the graph of y = 10x as the graph takes on x-values that are negative numbers having increasing absolute values.
X,θ,T If the graphs of y = 10x and y = log x were reflections in the line y = x, will the graph of y = log x ever touch the y-axis?
Three points that are known to be on the graph of y = 10x are (-1, 0.1), (0, 1), and (1, 10). Therefore, the points (0.1, -1), (1, 0), and (10, 1) should all be on the graph of y = log x. TRACE. Cursor down twice to access the graph of y = log x. The function is not defined for x = 0, so an error message appears. Cursor right to verify that the three points above are found.
Previously Known Now Established 10-1 = 0.1log 0.1 = -1 100 = 1log 1 = 0 101 = 10log 10 = 1 The logarithm (in base 10) of 0.1 is -1. The logarithm (in base 10) of 1 is 0. The logarithm (in base 10) of 10 is 1.
Previously Known Now Established 10 -1 = 0.1log 0.1 = -1 10 0 = 1log 1 = 0 10 1 = 10log 10 = 1 These are the logarithms (in base 10). What are logarithms, and what is the connection between exponential and logarithmic expressions?
4-2 = 0.0625log4 0.0625 4-1 = 0.25log4 0.25 40 = 1log4 1 41 = 4log4 4 42 = 16log4 16 Previously Known Predict These Verify the predictions: MENU; 1
Laws of Logarithms The calculator can be used to explore patterns involved in logarithmic arithmetic. These patterns will lead to the development of the three laws of logarithms. If the connection between parallel exponential and logarithmic expressions has been grasped, the laws of logarithms will not be surprising.