Unit 5: Exponential Functions, Logarithms, Sequences, and Series

Brittany Duncan, Bri Browning, Kaleb Jenkins, and Anna Blackshear

Exponentials • Exponential Function: a function with the

general form y=a, where x is a real number, a0, and b1.

• You can use an exponential function to model growth when b. When b, b is the growth factor.

Graph: X Y-3

-2

-1

0

1

2

3

Hint: you are going to have to change the y-axis to make it fit.

𝟏𝟖𝟏𝟒𝟏𝟐

2

1

4

8

Try it Yourself: y= 4(X Y

Logarithms• Logarithm: when

the base b of a positive number y is defined as follows: if y= , then = x.

• A logarithmic function is the inverse of an exponential function.

How to: Change Base

For any positive numbers, M, b, and c, with b1 and c1,

Example:Change to log base 3.

= 1.3869=1.3869= (1.3869=)0.6617= =xx4.5889

log 34.5889

Try One Yourself:Change to log base 8.

= 2.292=2.292=(2.292=)2.0699==x117.4627=x

log 8177.4627

Word Problems:A scientist carbon-dates a piece of fossilized

tree trunk that is thought to be over 5000 years old. The scientist determines that the sample contains 65% of the original amount of carbon-14. the half-life of carbon-14 is 5730 years. Is the reputed age of the tree correct? Explain.

SequencesArithmetic Sequences: the difference between consecutive terms is constant. (adding the same number every time)

Practice:Fund- Raising: Suppose you participate in a bike-a-thon for charity. The charity starts with $1100 in donations. Each participant must raise at least $35 in pledges. What is the minimum amount of money raised if there are 75 participants? Find the 76th term of the sequence 1100, 1135, 1170…

= + (explicit formula)

= 1100+ (76-1)(35)= 1100+(75)(35)=3725

With 75 participants, the bike-a-thon will raise a minimum of

$3725

SequencesGeometric SequencesThe ratio between consecutive terms is constant. (multiplying the same number for every number).

Practice:Design: Suppose you want a reduced copy of a photograph. The actual length of the photograph is 10 in. The smallest size the copier can make is 64% of the original. Find the length of the photograph after five reductions of 64%. For five reductions, you need to find the 6th term of the geometric sequence 10, 6.4,…= (explicit formula)

SeriesArithmetic Series a series whose terms form an arithmetic sequence. When you know only the 1st and last term of the sequence, you can use a formula to evaluate the related series quickly.

Example: Embroidery such as cross-stitching frequently decorates table linens and clothing, although it was originally used to strengthen and repair woven fabric. Several rows of cross-stitches make up the green roof. Find the total number of green cross stitches in the roof.Sum of the series is times (1st term plus last term)

=2.5(18)45

Summation Notation

You can use the summation symbol to write a series. Then you can use limits to indicate how many terms you are adding. Limited are the least and greatest integral values of n.

Explicit formula for

the sequence

Upper limit,

greatest

value of n.

Lower limit, least value of

n.

Try it yourself:Use Summation notation to write the series 3+6+9… for 33 terms.

∑𝑛=1

3

3𝑛

SeriesGeometric Seriesis the expression for the sum of the geometric sequence.

Example:

=1241.24