ALGEBRA I- MODULE 7Honors Topics

Exponential and Logarithmic Inequalities

• Exponential inequality rules:

• Logarithmic inequality rules:

If the bases of the exponential inequality are not the same, you must “log both side” to get the variable out of the exponent.

, then x yb b x y , then x yb b x y

If 1, then log 0bn n If 1, then log 0bn n

log log

log log

log log if 1 and if 1

log log

x

x

b c

b c

x b c

c cx b x b

b b

If log log , then b bn c n c If log log , then b bn c n c

**Always check solutions for logarithms- must have only positives after the log

Examples of Exponential and Logarithmic Inequalities

• Solve each inequality.2 1

2 1

1

1

3

0

3 x x

x x

x

x

log5.2 log 4

log5.2 log 4

log 4 You are by positive

log

5.

5.2

0.84 )

2 4

(

x

x

x

x

x approx

log 0.47 log8.1

log 0.47 log8.1

log8.1 You are by n

0

egativelog 0.

.47 8.

47

2.77( )

1x

x

x

x

x approx

3 3log ( 4) log (3 )

4 3

4 2

2 2

x x

x x

x

x x

2 2log (5 2) log ( 4

5 2 4

4 6

)

32

x x

x

x

x

x

Non-Arithmetic and Non-Geometric Sequences & Series

• We studied arithmetic and geometric sequences and series, but there are some sequences and series that are neither arithmetic nor geometric.

• Sequences can be generated using any pattern of n, the location and number of each term.

• generates the following terms. A table is a good way to organize the terms.

*This sequence does not have a common difference or common ratio

2 2na n

n 1 2 3 4 5 6

-1 2 7 14 23 34na2

3 3 2a 22 2 2a 2

1 1 2a 24 4 2a 2

5 5 2a 26 6 2a

Terms of Sequences• Find the first 4 terms of each sequence.

Terms: 0, 1/5, 1/3, 3/7

Terms: 5, 7, 11, 19

*These are all explicit formulas, but can you use recursive?

1

3n

na

n

n 1 2 3 4

0 1/5 1/3 3/71

3n

na

n

2 3nna n 1 2 3 4

5 7 11 192 3nna

Examples of Recursive Formulas• Find the first 4 terms of each sequence.

Terms: -4, -7, -13, -25

Terms: 5, 7, 11, 19

Now that you generated terms, can you write the formulas?

n 1 2 3 4

-4 -7 -13 -25na

2

1 1

1 given

2n na a a n 1 2 3 4

1/2 1/4 1/16 1/256

1 12 1 if 4n na a a

na

Write Explicit Formulas• You may want to organize the terms in a table to compare

the terms to the values of n.• Do you add to n? Subtract? Multiply? Divide? Square it?

• Write the explicit formula for the apparent nth term of the sequence.

• 1, 4, 7, 10, 13, …

Formula:

• 2, 5, 10, 17, 26

Formula:

n 1 2 3 4 5

1 4 7 10 13na3 2na n

n 1 2 3 4 5

2 5 10 17 26na2 1na n

Sigma Notation• Find the indicated sum.

4

1

1 1 1 1

1 2 3 412 6 4 3 25

12 1

1

2

i i

6

2

20 30 40 50 60

200

10k

k