Unit 5: Exponential & Logarithmic Functions · Web view5.1-5.75.1-5.7Unit 5: Exponential &...
Transcript of Unit 5: Exponential & Logarithmic Functions · Web view5.1-5.75.1-5.7Unit 5: Exponential &...
Unit 5: Exponential & Logarithmic Functions
2017
pebblebrook high schoolALGBRA 2
5.1-5.7
5.1 Graphing Exponential Functionsy = abx – c + h
Growth, b > 1 Decay, 0< b < 1
Starting Coordinate (0, a) Starting Coordinate (0, a)Domain: (−∞ ,∞¿ Domain: (−∞ ,∞¿Range: (h, ∞ ¿ Range: (h, ∞ ¿Vertical Asymptote y = h Vertical Asymptote y = h
Example #1: Identify a & b. Decide if the exponential function is a growth or decay.
a) y = 3(2)x
b) y = 12(3)x
c) y = 7(34 )x
d) y = 6(3)-x
Example #2: Describe the exponential function. Then, sketch the graph of the exponential function.
a) Y = 3(2)x
DescriptionStarting CoordinateVertical AsymptoteDomainRangeEnd Behavior
b) Y = 2(12)x- 3 + 4
c) Y = -4(3)x - 2DescriptionStarting CoordinateVertical AsymptoteDomainRangeEnd Behavior
You Try….
DescriptionStarting CoordinateVertical AsymptoteDomainRangeEnd Behavior
1) Y = 12 (2)2
DescriptionStarting CoordinateVertical AsymptoteDomainRangeEnd Behavior
2) Y = 8(12)x + 2 + 3
What happens when the exponential function is reflected on the axis?
Reference: 12 Basic Functions
DescriptionStarting CoordinateVertical AsymptoteDomainRangeEnd Behavior
Section 5.1 Homework
Identify a & b. Decide if the exponential is a growth or decay.
Describe the exponential function. Then, sketch the graph.
9. y = 8x + 5
10. y = 9(13)x + 7 – 3
5.2 Logarithmic Functions as Inverses
DescriptionStarting CoordinateVertical AsymptoteDomainRangeEnd Behavior
DescriptionStarting CoordinateVertical AsymptoteDomainRangeEnd Behavior
Example #1: Write the exponential in Log Form.
a) 53 = 125
b) 42 = 16
c) (-3)4 = 81
d) gx = h
e) a3 = x – 2
Example #2: Write the log function in exponential form.
a) log2 8 = 3
b) logr y = q
c) log3 (x – 2) = 4
Your calculator ONLY computes on base 10. To perform calculations in any other base you must you the Change of Base.
Logb y = log ylogb
Example #2: Evaluate.
1) log8 16 =
2) log 400 =
3) log3 24 =
You Try…
1) Write in log form: 64 = 1296
2) Write in exponential form: log5 3125 = 5
3) Evaluate: log8 54 = ?
Section 5.2 Homework
5.3 Properties of Logarithmic Functions
Example #1: Identify the properties of Logs
Example #2: Write as a single log.
Examples #3: Expand the log expression.
Section 5.3 Homework
Identify the log property.
Write as a single log expression.
Expand the log expression.
5.4 Solving Exponential Equations
Steps for solving exponential equations
Simplify, if necessary. Rewrite in Log Form.
Undo additions/subtractions, if necessary. Undo multiplications/divisions, if necessary.
Use the change of base form. Solve the equation for x.
Example: Solve the exponential equation.
1. 73x = 20
2. 8 + 10x = 1008
3. 5x + 1 = 24
4. 72x – 1 = 371
Section 5.4 Homework
Section 5.5 Graphing Logs & Natural Logs
Y = h + a log (x – c)
Important Parts: Starting coordinate: (a, 0)
Domain: (h, ∞) Range: (-∞ ,∞)
Horizontal Asymptote: x = c
Example #1: Describe the transformation of the log function. Sketch the graph.
1) Y = log2 x
2) Y = 3 + log6 (x – 2)
DescriptionStarting CoordinateHorizontal AsymptoteDomainRangeEnd Behavior
Important Parts:
Starting coordinate: (a, 0) Domain: (h, ∞) Range: (-∞ ,∞)
Horizontal Asymptote: x = c
Example #2: Describe the transformation of the natural log function. Sketch the graph.
1) Y = 3 + ln (x +2)DescriptionStarting CoordinateHorizontal AsymptoteDomainRange
Remember, finding inverses… Switch x & y, rewrite in log/ln form, & solve for y.
Example #3: Find the inverse.a) y = log7 24x
DescriptionStarting CoordinateHorizontal AsymptoteDomainRangeEnd Behavior
b) y = ln x - 2
c) y = log2 (x - 1)
Section 5.5 Homework
13. Find the inversea) y = Log2 6x
b) y = Ln x + 5
c) y = log4 (x - 3)
Section 5.6 Solving Log equations
Steps for Solving Log equations
Write the log as one expression Isolate the log
Rewrite in exponential form. Solve for x.
Examples: Solve the log equation.
1) log (3x + 1) = 5
2) 2 log x + log 3 = 2
3) log (6x) – 3 = -4
4) log (5 – 2x) = log (-5 + 3x)
5) log (7x + 1) = log (x – 2) + 1
Section 5.6 Homework
Section 5.7 Natural Logs
Example #1: Simplify the natural log.
1) 3 ln 6 – ln 8
2) ln 9 + ln 2
3) 2 ln 8 – 3 ln 4
Example #2: Expand the natural log.
1) ln ¿)2
2) ln(2m3n)
3) ln a2b3
c
Example #3: Solve the natural log equation.
1) ln (3x + 5)2 = 4
2) ln (x – 1) = 3
3) 3 ln x + ln 5 = 7
Section 5.7 HomeworkCondense.
Expand.
Solve.