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MATH 3 - MODULE 6Honors 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 0 < 1, then log 0bn n
log log
log log
log log if 1 and if 0 < 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