10-1: Exponential Functions – Day One Ex. 1: On the axis below, sketch the graphs of: a) 2 b) 3 c) 5 What point(s) do all of these graphs have in common? What appears to be the domain of each graph? What appears to be the range of each graph?
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10-1: Exponential Functions – Day One Ex. 1: On the axis below, sketch the graphs of:
a) � � 2� b) � � 3� c) � � 5�
What point(s) do all of these graphs have in common? What appears to be the domain of each graph? What appears to be the range of each graph?
In general, any function of the form ______________________, is called an ____________________________ ___________________________________ with base ________. Exponential functions have the following characteristics:
1) The function is ______________________________ and one-to-one.
2) The domain is the set of _________________________________________.
3) The �-axis is an ______________________________ of the graph.
4) The range is the set of all _____________________ numbers if � 0 and all _____________________ numbers if � � 0.
5) The graph contains the point ___________. That is, the �-intercept is _________.
6) The graphs of � � ��� and � � �� � are reflections across the _________________________. Ex. 2: Using the graphing calculator, sketch graphs of � � 2� and � � 2 �. These are examples of two types of exponential functions: __________________ and __________________.
_________________________________________________________
__________________________________________________________
__________________________________________________________ Exponential functions are frequently used to model ___________________ or ____________________ of populations.
HW: Day One
10-1: Exponential Functions Con’t.
If 2� � 8 and 2� � 8, what does � equal?
HW: Day Two
10-2: Logarithms and Logarithmic Functions – Day One
Fill in the following tables of values and then draw both graphs of the coordinate axis.
� � 2�
� �
�3
�2
�1
0
1
2
3
� � 2�
� �
�3
�2
�1
0
1
2
3
Note that the graphs of the two functions are reflections of each other over the line __________. In general, the inverse of � � �� is � � ��. In � � ��, � is called the ___________________________ of �. It is usually written as _____________________________________________ and is read ____________________________________________________________________. ***************So, � � log� � if and only if � �___________.********************
HW: Day One
10-2: Logarithms and Logarithmic Functions – Day Two Sketch a graph of � � log� �.
The graph shown above is an example of a ___________________________________ function. Logarithmic functions have the following characteristics:
1) The function is _______________________________________- and one-to-one.
2) The domain is the set of ____________________________________________________________.
3) The ___________________ is an asymptote of the graph.
4) The range is the set of ______________________________________________________________.
5) The graph contains the point �1,0�. That is, the �-intercept is _________________. Since exponential functions and logarithmic functions are inverses, they “undo” each other.
HW: Day Two
Prove if the statement is true or false.
10-3: Properties of Logarithms Using your calculator, evaluate the following:
a) log 10 �
b) log 5 � log 2 �
c) log 18 �
d) log 6 � log 3 � What rules do these examples suggest? log ��� �__________________________________________________
Using your calculator, evaluate the following:
a) log 50 � log 2 �
b) log 25 �
c) log 100 �
d) log 500 � log 5 � What rules do these examples suggest? log �
��_____________________________
Using your calculator, evaluate the following:
a) log 25 �
b) log 5�� �
c) 2 log 5 �
d) log 64 �
e) log 8�� �
f) 2 log 8� �
What rules do these examples suggest? log ��� � ______________________________________________.
HW:
10-4: Common Logarithms The base 10 is one of the two most important bases in logarithms. Base 10 is so important that it gets its own name: ____________________________________. When we write the common log, we usually only write ___________________________________________.
The inverse of the common log would be __________________________________________________.
Is there a way to evaluate ALL logs using the common log? _____________________________________.
HW:
10-5: Base � and the Natural Log – Day One Using your calculator, fill in the following table:
� � � �1 � 1��
�
1 �1 � 1
1�
4
12
365
8760
86,400
100,000
As our values for � increase, what is happening to �? Definition: The natural base � is the number such that � � ________________________________________. � is an __________________________________ number, like �. An exponential function with base � is called a __________________________________ exponential function. Sketch the graph of � � �� below.
a) �� b) ��.�
Ex. 1A: Simplify the following without using a calculator.
a) �� � �� � b) �� � �� �
c) ��� � �� � d) � �!
Since there is an exponential function with base �, there must also be a ____________________________ function with base �. The logarithm with base � is called the _____________________________________. The symbol for the natural logarithm is ___________________________________________. The natural logarithmic function is written ______________________________________. Sketch graphs of � � �� and � � ln $ on the axis below.
Since the natural base and the natural log are inverses, they _________________________ each other out.
HW: Day One
Section 10-5: Day Two
Ex. 8: Solve the following:
a) b) c)
d) e) f)
HW: Day Two
Section 10-6: Exponential Growth and Decay What happens to the price of a car after you buy it and drive it off of the car lot? Suppose a brand new Cadillac Escalade costs $54,000. Research has shown that it loses, on average, 23% of its value every year. Fill in the following table to find out the value of a 2012 Cadillac Escalade in 5 years. Year Value at Start of Year Formula Value at End of Year
2012 54,000
2013
2014
2015
2016
The ___________________________ of the value of a car is an example of exponential _________________. When a quantity ___________________________ by a fixed __________________ each year, or any other period of time, the amount � of that quantity is after � years is given by _______________________________. � is the initial amount, and � is the percent of decrease expressed as a decimal. The percent of decrease �, is also referred to as the _____________________________________.
When scientists use exponential decay, they prefer to use the equation:________________________________
When a quantity _____________________________ by a fixed percent, it is no longer called exponential decay, it is now called exponential _______________________. � is now called the ___________________________________________.
HW:
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