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.