and
= log648 = log1288
log648 = x log1288 = x
64x = 8 128x = 8
26x = 23 27x = 23
6x = 3 7x = 3
The next two terms for the second sequence is based off this equation: where n represents the term number. In this case the next two term numbers are 5 and 6.
and
= log24381 = log72981
log24381 = x log72981 = x
243x = 81 729x = 81
35x = 34 36x = 34
5x = 4 6x = 4
The next two terms for the third sequence is based off this equation: where n represents the term number. In this case the next two term numbers are 5 and 6.
and
= log312525 = log1562525
log312525 = x log1562525 = x
3125x = 25 15625x = 25
55x = 52 56x = 52
5x = 2 6x = 2
Write the nth term of each of these sequences, in logarithmic form and in the form . Explain how you got these values.
The nth term of the first sequence is:
In the form p/q:
2nx = 23
nx = 3
The nth term of the second sequence is:
In the form p/q:
3nx = 34
nx = 4
The nth term of the third sequence is:
In the form p/q:
5nx = 52
nx = 2
Now consider the general sequence,
Determine the values of the first five terms. Explain how you got these values.
logmmk = x
mx = mk m2x = mk m3x = mk m4x = mk m5x = mk
x = k 2x = k 3x = k 4x = k 5x = k
Write the nth term of this sequence, in logarithmic form and in the form . Explain how you got this value.
The nth term of this sequence is:
mnx = mk
nx = k
What must be the relationship between the argument and first base if each term in the sequence is to have the form ?
The relationship is that both the sequences have the form where mq equals the base and mp equals the resultant number of the logarithm. When both the base and resultant number are turned into the same base m to the power of p and q, the answer will be in the form of .
PART 2 Exploring
Determine the numerical values of the following sequences. Explain how you got these values. Justify your answers using technology.
Use of Technology
log464 = x log864 = x log3264 = x
4x = 64 8x = 64 32x = 64
4x = 43 23x = 43 25x = 43
22x = (22)3 23x = (22)3 25x = (22)3
2x = 6 3x = 6 5x = 6
x = 3 x = 2
log749 = x log4949 = x log34349 = x
7x = 49 49x = 49 343x = 49
7x = 72 72x = 72 73x = 72
x = 2 2x = 2 3x = 2
x = 1
5-3x = 53 5-x = 53 5-4x = 53
-x = 3 -3x = 3 -4x = 3
x = -3 x = -1
log8512 = x log2512 = x log16512 = x
8x = 512 2x = 512 16x = 512
23x = 83 2x = 83 24x = 83
23x = (23)3 2x = (23)3 24x = (23)3
3x = 9 x = 9 4x = 9
x = 3
The third answer in each row can be obtained from the first two answers in that row. Explore several of these examples to conjecture a way to combine the first two answers to get the third. Confirm this conjecture in the remaining examples. Create two more examples to test your conjecture.
The formula that combines the first two answers to get the third is: (ab)/(a+b) where a is the first answer and b is the second answer of the sequence.
Proof:
First Sequence Second Sequence Third Sequence
First answer: 3 First answer: 2 First answer: -3
Second answer: 2 Second answer: 1 Second answer: -1
Third answer: Third answer: Third answer:
Fourth Sequence
First answer: 3
Second answer: 9
Third answer:
Two more examples:
log3 81, log9 81, log27 81
log3 81 = x log981 = x log2781 = x
3x = 81 9x = 81 27x = 81
3x = 34 32x = 34 33x = 34
x = 4 2x = 4 3x = 4
x = 2
First answer: 4
Second answer: 2
Third answer:
log6 216, log36 216, log216 216
log6216 = x log36216 = x log216216 = x
6x = 216 36x = 216 216x = 216
6x = 63 62x = 63 63x = 63
x = 3 2x = 3 3x = 3
x = 1
First answer: 3
Second answer:
Third answer: 1
Now consider the general case of .
Let
Determine the general equation for in terms of and .
The general equation for logab x in terms of c and d is:
Proof:
logax = c, ac = x, a =
logbx = d, bd = x, b =
logabx = y
logabx = y
(ab)y = x
Test the validity of this equation using other values of , , and .
Testing the equation:
log11 1331, log121 1331, log13311331
log111331 = x log1211331 = x log13311331 = x
11x = 1331 121x = 1331 1331x = 1331
11x = 113 112x = 113 113x = 113
x = 3 2x = 3 3x = 3
x = 1
First answer: 3
Second answer: 3/2
Third answer: 1
Discuss the scope/limitations of , , and .
Limitations:
a, b, ab:
- has to be always bigger than 0 but not equal to 1 because we cannot evaluate the logarithm of a negative base
-
cannot equal to 1. Using the change of base formula, the logarithm of the bases is the denominator of the equation,. When y equals to 1, the denominator is 0 as the logarithm of 1 is 0. Any number divided by 0 is undefined.
x:
- is bigger than 0 because we cannot evaluate the logarithm of a negative number
Use the rules of logarithms to justify this equation.
Given: logax = c, logbx = d, logabx = ?
(equation 1)
(equation 2)
(from equation 1 and 2)
