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TRANSCRIPT
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Table of ContentsClick on the topic to go to that section
SlopesEquations of LinesFunctionsGraphing FunctionsPiecewise FunctionsFunction CompositionFunction RootsDomain and RangeInverse FunctionsTrigonometryExponentsLogs and Exponential Functions
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Recall from Algebra, The SLOPE of a line is the ratio of the vertical movement to the horizontal movement. In other words, it describes both the steepness and
direction of a line.
Slope
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One way to determine the slope is calculate it from two points.
Consider two points, (x1,y1) and (x2,y2)
The slope, m, is:
*Note: a slope is not defined for a vertical line (where x1=x2)
Calculating Slope
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Example: Calculate the slope of the line containing the points
(3,4) and (2,8)
Calculating Slope
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1 What is the slope of the line containing the points:
(15,-7) and (3,5) ?
A m= 1
B m= -1
C m= -1/11
D m= 2/8
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2 What is the slope of the line containing the points:
(2,2) and (8,3) ?
A m= 6
B m= 5
C m= 1/2
D m= 1/6
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3 What is the slope of the line containing the points:
(17,23) and (-6,-18) ?
A m= 41/23
B m= 23/41
C m= -2
D m= -23/41
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Once you have the slope of a line, it is important to be able to write the equation for the line.
If you have the slope of the line, m, and any one point, (x1, y1), you can write the equation of the line.
Let be a point, then
This form is called Point-Slope Form of an equation. Point-Slope Form is extremely useful in Calculus and it is important that you are comfortable using it.
Point-Slope Form
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Example: Find the equation of the line that has a slope of 4 and passes through the point (-2, 5). Write the answer in Point-Slope form.
Point-Slope Form
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- Write the equation for the line in point-slope form, that has a slope of 4 and contains the point (5,-8).
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- Write the equation of the line, in point-slope form, that has a slope of -5 and contains the point (3,15).
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- Write the equation of the line, in point-slope form, that contains the points (5,3) and (-3,-6).
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- Write the equation of the line, in point-slope form, that contains the points (-4,3) and (2,9).
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Recall from Algebra, another common way to express the equation of a line is called slope-intercept form.
This is written as:
Where m is the slope, and the y-intercept is at (0,b).
Slope-Intercept Form
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Example: Find the equation of the line with a slope of 3, containing the point (4,5). Express your answer in slope-intercept form.
Slope-Intercept Form
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- Write the equation of the line, in slope-intercept form, that has a slope of 5 and contains the point (23,15).
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- Write the equation of the line, in slope-intercept form, that has a slope of -3 and contains the point (6,8).
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- Write the equation of the line, in slope-intercept form, that contains the points (16,14) and (-2,-7).
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A function is a relationship between x and y such that for any value x , there will be one and only one value of y.
For example:
1.2.
What is a Function?
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For the function definition given on the previous slide to be true, the function will also pass what is called the Vertical Line Test. This states that a graph is of a function if and only if there is no vertical line that crosses the graph more than once.
For the same examples, let's look at their graphs:
1. 2.
Vertical Line Test
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Here is an example of how that would be expressed:
x y1 63 11
12 557 9
There is no given equation for this relation, but it is a function since there is only one y value for each x value.
A third way to demonstrate functions is in tabular form. Sometimes functions can be represented as a set of ordered pairs, or a relation. This is used often when the equation itself is unknown.
Functions as a Table
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Sometimes it is useful to consider relations that are not functions.
If for any input there is more than one output, it is not a function. Here are examples of equations that are not functions:
1.2.
Equations Which are Not Functions
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You can see that both examples do not pass the Vertical Line Test:
1. 2.
Failing the Vertical Line Test
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6 Is the following relation a function?
Yes
No
-103
-2-1 0
x y
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7 Is the following relation a function?
Yes
No
-2 3
0 2
-1 -1
3 2
4 0
Ans
wer
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10 What is the value of f(x+2) given
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It is important to be able to graph functions. At this point, you should be familiar with methods for doing so. You should also be
able to understand parent graphs, and identify shapes and orientations of different, common functions.
Graphing Functions
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y = a f( bx ∓ c) ± d
Transforming FunctionsFunctions, like equations, are transformed in a predictable manner. Each letter below has a separate effect on a given function. Identify how each letter transforms a function.
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11 Which of the following is the graph of ?
A B
C D
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12 Which of the following is the graph of ?
A B
C D
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13 Which of the following is the graph of ?
A B
C D
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14 Which of the following is a graph of ?
A B
C D
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15 Which of the following is a graph of ?
A B
C D
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16 Which of the following is the graph of ?
A B
C D
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From the previous slide's question, see if you can write the equations for the other graphs:
B C D
Further Challenge
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17 Which of the following is a graph of ?
A B
C D
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From the previous question, see if you can you write the equations for the three other graphs:
A B C
Further Challenge
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Piecewise functions can be thought of as several functions at once, each defined on a specific interval, or each used in a different region.
To graph a piecewise function you do not plot the entire graph of each individual section - graph only the parts defined by x.
Piecewise Functions
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A simple example of a piecewise function is the absolute value function.
The graph of this function looks like this:
Note, that at the point x=0, the two function pieces meet. This is not always the case.
Piecewise Functions
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Some piecewise functions can be discontinuous. When you have a piecewise function in which the different sections do not meet, there is special notation for the end points.
included endpoint/solid circle
discluded endpoint/open circle
Discontinuity Notation
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Example: Evaluate the following piecewise function at the given points:
Evaluating a piecewise function is the same as a continuous function, however we must pay close attention to the endpoint definitions.
Evaluating Piecewise Functions
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Now we can practice graphing the same piecewise function.
Graphing Piecewise Functions
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Example: Evaluate the piecewise function at the given values:
Evaluating Piecewise Functions
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Example: Graph the following piecewise function:
Graphing Piecewise Functions
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- Given the following piecewise function, find the value of
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- Given the following piecewise function, find the value of
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- Given the following piecewise function, find the value of
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To evaluate composite functions, you must start from the innermost "layer" and work your way out.
For example, if and
To evaluate , first x passes through the function g(x), and that output is then plugged into f(x).
Evaluating Composite Functions
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Example: Given and find
Evaluating Composite Functions
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22 What is the value of given the following functions:
A
B
C
D
145
26-4
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23 What is the value of given the following functions:
A
B
C
D
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24 What is the value of given the following functions:
A
B
C
D
15
77
197
152
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25 Find the value of
A
B
C
D
11
9
-3
15
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26 Given and , find h(x) if
A
B
C
D
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28 Given and , find h(x) if
A
B
C
D
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29 Given and , find h(x) if
A
B
C
D
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Another important idea to understand regarding functions is the roots of the function.
A root, sometimes called a zero solution of f(x), is the value of x such that f(x)=0. It can also be called the x-intercept.
roots/zeroes/x-intercepts
Roots of a Function
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Example:Find the roots of the following:
One method for finding roots is to factor and use the zero product property. For quadratics that are unfactorable, the quadratic formula can be used.
Calculating Roots
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Sometimes the equations are not as easily factorable, and the quadratic formula is required.
Recall: ;
Example: Find the roots of the following equation:
Quadratic Formula
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Recall from Algebra II, the Domain of a function is the set of all possible inputs for a function, typically the x-values.
Similarly, the Range of a function is the set of all possible outputs for a function, typically the y-values.
Domain and Range
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Watch for values which may cause: –zero in the denominator –square roots of negative numbers – logs of zero – logs of negative numbers
Certain conditions must be avoided in order for the Domains and Ranges of functions to be real.
Domain and Range
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Example: Find the Domain and Range of the following function:
Domain and Range
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Example: Find the Domain and Range of the following function:
Domain and Range
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33 What is the Domain and Range for the following function:
A
B
C
D
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34 What is the Domain and Range of the following function:
A
B
C
D
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35 What is the Domain and Range for the following function:
A
B
C
D
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Sometimes more complicated functions are presented. In this case, finding ranges might be very difficult, and finding domains are more important.
Example: find the Domain for the following function:
A More Challenging Example
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36 What is the Domain (only) for the following function:
A
B
C
Domain: All real numbers
Domain: All real numbers except x=-3, x=2 and x=-5
Domain: All real numbers except x=-3 and x=-5
Domain: All real numbers except x=3 and x=5D
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37 What is the Domain (only) for the following function:
A
B
C
D
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In order to study inverse functions, it is first necessary to specify which kind of functions are appropriate.
We know that for a relation to be a function, every value in the domain must have exactly one value in the range. For a function to have an inverse, we further require that every value in the range must have exactly one value in the domain.
In other words, no two values of x yield the same y.
This relationship is called a One-to-One Function.
One-to-One Functions
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You must determine if a function is One-to-One, in order for you to then find it's inverse.
If given ordered pairs, simply look to see if there are no repeated y-values.
If given an equation that is easy to plot, you can use the Horizontal Line Test. This states that if it is possible to draw a Horizontal line anywhere on the graph, and it crosses the graph more than once, it fails the One-to-One requirement.
Horizontal Line Test
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Example:
Notice: The line crosses the graph twice and fails the Horizontal Line Test. Therefore, it is not a One-to-One function.
Failing the Horizontal Line Test
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Example:
Notice: The line does not cross the graph more than once and Passes the Horizontal Line Test. Therefore it is a One-to-One function.
Passing the Horizontal Line Test
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38 Is the following graph a One-to-One function?
Yes
No
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39 Is the following graph a One-to-One function?
Yes
No
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Example: Find the inverse of f(x), given:
Finding the Inverse
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Step 5 involves the previously discussed Function Composition. (click for link)
Inverse Function can be defined as:
andGiven two One-to-One Functions
if: and
then and are Inverses of each other.
Inverse Definition
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Example: Given:
Are these two functions inverses of each other? Check to make sure it follows the definition.
Inverses
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The Inverse of is
and the Inverse of is
Terminology
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40 Which of the following is the correct notation for the Inverse Function of ?
A
B
C
D
E
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41 Given the following function, which is its inverse function?
A
B
C
E Not Invertable
D
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42 Given , Find
A
B
C
E Not Invertable
D
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43 Given , Find
A
B
C
E Not Invertable
D
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Another special relationship that you may recall about functions and their inverses is that their graphs are a reflection across the line y=x.
Graphs of Inverses
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Trig FunctionsThese are the six trig functions you are familiar with from Geometry and Precalculus.
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All these trig functions are defined in terms of a right triangle:
OppositeHypotenuse
Adjacent
Trig - Right Triangles
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The graphs of these functions should be easily recognizable:
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The range of sin and cos ?
The range of csc and sec ?
The range of tan and cot ?
The ranges for these functions can also be determined.What is:
Range of Trig Functions
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Another important matter is the sign of the trig functions in each quadrant. The letters A-S-T-C represent the positive values. All other trig functions will be negative in those quadrants.
A: All trig functions are positive in the 1st quadrant.
S: Sin values are positive in the 2nd quadrant.
T: Tan values are positive in the 3rd quadrant.
C: Cos values are positive in the 4th quadrant.
A-S-T-C
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In Calculus class almost all problems are in radians, not in degrees. This table shows the "special" angles, in both, that you should be familiar with.
Degrees 0 30 45 60 90 180 270 360
Radians 0
Radians
Teac
her N
otes
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In Geometry and Pre-calculus you learned quite a bit about trigonometry. To be successful in calculus, it is
very important that you know how to evaluate trig functions at various angles. Many real life situations behave in a trigonometric pattern (i.e. traffic flow), therefore you will see that trig functions are very
prevalent in the course and on the AP Exam.
Trigonometry
Teac
her N
otes
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1. THE UNIT CIRCLE
This method requires you to memorize values for each ordered pair. Recall that the x value of each ordered pair is the cosine value, while the y value of the ordered pair is the sine value.
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The Unit Circle is divided into 4 quadrants. They are listed below.
III
III IV
The Unit Circle
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The x and y coordinates for special angles in the other quadrants can be determined by knowing the similar 1st quadrant angle's value. The x and y values will be the same, but the signs will (or can) be different.
Special Angles in the II, III, and IV Quadrants
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This method requires you memorize values from the table and remember:
2. THE TRIG TABLE
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This method requires you to draw any of the above triangles on a set of axes depending on given angle, and remember:
3. SPECIAL RIGHT TRIANGLES
Teac
her N
otes
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46 Evaluate
A B C D E
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50 Evaluate
A
B
C
D
E
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51 Evaluate
A
B
C
D
E
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Double Angle Formulas
The following Trig Identities are some of the more common ones, you may recall from Pre-calculus.
Pythagorean Identity
Half Angle Formulas
Sum Identities
Trig Identities
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52 Evaluate
A
B
C
D
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Inverse Trig FunctionsInverse Trig Functions follow the same rules as other Inverse Functions we learned earlier. (Click here)
They "undo" what the trig function does. For example if the function is then the inverse trig function is .
You may also see the following terminology.
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For sinx:
For cosx:
For tanx:
Remember that Inverse Functions must be One-to-One. Recalling our basic trig graphs, we can see that none of them are One-to-One. Therefore, we must restrict the range.
(Click here)
Inverse Functions
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Example: Evaluate
In other words, we must find what angles have sin values of , remembering our range restrictions.
Evaluating
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56 Evaluate
A
B
C
D
E
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57 Evaluate
A
B
C
D
E
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Properties of Exponents
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Simplify each of the following expressions.
Practice
Ans
wer
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59 Simplify.
A
B
C
D
E None of the above
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60 Simplify:
A
B
C
D
E None of the above
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63 Simplify:
A
B
C
D
E None of the above
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65 Simplify:
A
B
C
D
E None of the above
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69 Find
A
B
C
D
E
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Log Properties:
Change of Base formula:
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Example: Find
Logarithms
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70 Find
A
B
C
D
E
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71 Find
A
B
C
D
E
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72 Find
A
B
C
D
E Undetermined
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73 Find
A
B
C
D
E
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Special Case of Log
This is called the natural log, and it has a base of . follows the same rules and has the same properties as .
Note that:
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Using what we learned about the relationships between logs and exponents, we can now solve equations containing them.
Exponential and Logarithm Equations
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Example: Solve for x: (remember domain requirements for log)
Exponential and Logarithm Equations
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78 Solve for x:
A
B
C
D
E None of the above