5th Log 32 8 = .6 3/5 = .6
6th Log 64 8 = .5 3/6 = .5
7th Log 128 8 = .43 3/7 = .43
Find an expression for the nth term of each sequence. Write your expressions in the form P/Q.
Log 3 81 , Log 9 81 , Log 27 81 , Log 81 81 , Log 243 81 , Log 729 81
This sequence can be expressed 4/N. N being the nth term of the sequence.
1st Log 3 81 = 4 4/1 = 1
2nd Log 9 81 = 2 4/2 = 2
3rd Log 27 81 = 1.33 4/3 = 1.33
4th Log 81 81 = 1 4/4 = 1
5th Log 243 81 = .8 4/5 = .8
6th Log 729 81 = .66 4/6 = .66
Find an expression for the nth term of each sequence. Write your expressions in the form P/Q.
Log 5 25 , Log 25 25, Log 125 25 , Log 625 25, Log 3125 25 , Log 15625 25
This sequence can be expressed 2/N. N being the nth term of the sequence.
1st Log 5 25 = 4 2/1 = 2
2nd Log 25 25 = 2 2/2 = 1
3rd Log 125 25 = 1.33 2/3 = .66
4th Log 625 25 = 1 2/4 = .5
5th Log 3125 25 = .8 2/5 = .4
6th Log 15625 25 = .66 2/6 = .33
Now calculate the following. Giving your answers in the form of P/Q.
Log 4 64 = 3, Log 8 64 = 2, Log 32 64 = 6/5
Log 7 49 = 2, Log 49 49 = 1, Log 343 49 = 2/3
Log 1/5 125 = -3, Log 1/125 125 = -1, Log 1/625 125 = -3/4
Log 8 512 = 3, Log 2 512 = 9, Log 16 512 = 9/4
To obtain the third answer from each row using only the first two answers, you must first multiply the first and second answer together and then divide by the sum of the first and second answers.
Two more examples that fit the pattern are listed below:
Log 2 4 = 2, Log 4 4 = 1, Log 8 4 = 2/3 (2)(1)/(2 + 1) = 2/3
Log 2 1024 = 10, Log 4 1024 = 5, Log 8 1024 = 3.33 (10)(5)/(10 + 5) = 3.33
Let Log a x = c and Log b x = d. Find the general statement that expresses Log ab x, in terms of c and d.
If we do this than the general statement that expresses this is (c)(d)/(c + d).
For example let:
A = 2 C = 4
B = 8 D = 1.33
X = 16
Log 2 16 = 4 = C Log 8 16 = 1.33 = d Log 16 16 = 1
(4)(1.33)/(4 + 1.33) = 5.33/5.33 = 1
Discuss the scope and/ or limitations of a, b, and x.
Some basic restrictions for the general statement are that both a and b must be greater than 0. Logs have to have positive numbers as bases so a > 0, b > 0.
Log -5 25 = Nonreal Answer
Another limitation of a and b is that (a)(b) cannot equal 1.
Log 4 16 = 2, Log 1/4 16 = -2, Log 1 16 = ?
1 raised to the power of any number will always be 1 so it can never equal 16 thus the equation is flawed when (a)(b) = 1
(2)(-2)/(2 + (-2)) = -4/0
Using the general statement only further proves how (a)(b) cannot = 1 because it would cause you to divide by 0 which we are not able to do.
The general statement that I applied throughout the assessment (c)(d)/(c + d) I arrived at through plugging numbers in. Through a system of trial and error I was able to figure out the pattern and discover how you can deduce the third answer from the first two in the sequence using logs. Of course the general statement only applies to sequences of logs that follow the pattern of Log a X = c, Log b X = d , Log ab X = (c)(d)/(c + d).
