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Sequence 2: Log 3 81, Log 9 81, Log 27 81, Log 81 81, Log 243 81, Log 729 81…
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Sequence 3: Log 5 25, Log 25 25, Log 125 25, Log 625 25, Log 3125 25, Log 15625 25…
I noticed that all have numerator as 3 in p/q in sequence 1. The nth term is 3/ nth. Similarly, sequence 2 have common numerator as 4. So the nth term is 4/ nth in sequence 2. Lastly, Sequence 3 had common numerator 2, which allowed me to guess that 2/ nth is nth term for sequence 3. If this pattern is true, I can state that in sequence 4, the nth term in p/q expression is Log m mk / nth.
Part III & IV
In part three, I will calculate the given logarithms in the form p/q, where p, q∈ℤ. I will again use TI- 83 calculator to solve and double check the answer.
Log 2 64, Log 8 64, Log 4 64
Log 4 1024, Log 32 1024, Log 8 1024
Log 7 343, Log 1/7 343, Log 1/49 343
Log 1/5 125, Log 1/25 125, Log 1/125 125
Log 2 512, Log 8 512, Log 4 512
From the first table above, I noticed at first that you divide first two logarithms to get the third answer. If the first and second equation is divided, it will looks like:
Log 64 X Log 8 Log 64 X Log 8
Log 2 X Log 64 Log 2 X Log 64
The Log 64 cancel each other out and only Log 8/ Log 2 is left. If you put Log 8/ Log 2 into your calculator, the answer is 3, which is third logarithm’s answer in p/q form. However, rest of 4 sequences did not worked like the first sequence. I was able to see the clear connections between the bases of logarithms, but failed to found the relationship between 1st and 2nd to come up with the 3rd logarithm. I found out that bases are closely related, and was able to establish the pattern just for the base: 2nd base divided by 1st base will give 3rd logarithm’s base in all 5 sets. Also it had the same value m…
So then I substituted the 1st and 2nd logarithm as a and b respectively. I tried to multiply, subtract, add, and combination of these method. Finally, when I tried multiplication, division and subtraction all together, the fitting pattern appeared. The pattern I found is like this:
*a= 1st logarithm, b= 2nd logarithm*
ab = 3rd logarithm
(a-b)
For example, I used 2nd set of logarithms, which is Log 4 1024, Log 32 1024, Log 8 1024. If I substitute first and second logarithm with a and b respectively, I will get following numbers: (5)(2) = 10
(5-2) 3
Here, I can see that it gives me the 3rd answer in form of p/q form. So I ended up with two distinctive patterns. With this two patterns I found, I will now try to make same kind of sets see if it follows the rules when I place different numbers in the base.
Part V
Here in part five, I’ll try to create two more examples that fit the two patterns from Part IV within of my knowledge of understanding it. I’ll use calculator to get the answer that has fraction, and it is rational.
Example 1: Log 1/9 729, Log 1/729 729, Log 1/81 729
Example 2: Log 6 216 Log 216 216, Log 36 216
It seems that these two examples fit on all two patterns. Both sets pass the relationship of bases and the equation I discovered during the investigation.
Part VI
Now, I will try to find general statement that expresses Log b/a X in terms of c (Log a X) and d (Log b X). There is really no way to do this in other ways other than algebraically, so I will write steps. First I changed all logarithms into exponents. Then I made expression Log b/a X into an equation: Log b/a X = y
- Find y……
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(b/a)y= x = ac
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(b/a)y= ac
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(by)(1/a) y= ac ← stretching the equation
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(a(c/d)y)(1/a) y= ac ← substituted b to a(c/d)
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a(cy/d)(a)-y= ac ← change all expressions with base a
- (cy/d)-y = c
- cy-dy = cd
- y(c-d) = cd
- y = cd/(c-d)
As it turns out, the equation I found here was equal to what I found in part IV! Also, the relationship between the bases that I found was also in it, here represented by c and d. I think it is not a coincident that I found those two patterns in both parts.
Part VII
In this section, I will put different types of integers in a b or x to see this equation is valid. I will use the TI- 83 plus calculator to find exact value or close to 3 significant figures.
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Example 1: Log 1/100 1000, Log 1/1000 1000, Log 1/10 1000
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Example 2: Log 2 216 Log 8 216, Log 4 216
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Example 3: Log 2 1 Log 8 1, Log 4 1 Example 5: Log 2 -8 Log 8 -8, Log 4 -8
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Example 4: Log 2 0 Log 8 0, Log 4 0
It fits the equation y = cd/(c-d). Also I see the expression Log b/a X as well. I double checked with the calculator and it is valid. I seems like if the x is same for all three expressions, and have bases that 2nd /1st = 3rd equation, the general statement Log b/a X = cd/(c-d) is true. However if any of these are incorrect, the expression no longer works. There are some limitations on x, a, and b.
Part VIII
Now, since I found the general expression of Log b/a X, I will attempt to look for the limitation of a, b, and x by studying the precious parts.
As I analyzed the precious parts, I found that a, and b’s range is (0, 1)U(1, ∞), range for the base of the logarithm. If a, or b = 1 or a, or b ≤ 0, this expression will not work. Also, if x is not same in all three logarithms in the set, this statement will no longer work. Finally, if x is 1, 0, or negative integers, this expression no longer works as you see in example 3, 4 and 5.
Part IX (how did I come this far?)
At first, I was given with the set of logarithms. I analyzed them and put them into form of p/q. At the part II, I found the sets’ p/q form follows the expression Log m mk / nth. So In part 3 and 4, I tried to find a pattern that was in the given sets of logarithms. It took some time, but I found out that three logarithms had same x and 3rd Log’s base was product from 2nd Log’s base being divided by 1st Log’s base.
After discovery, I attempted to find a pattern for the set by substituting first two Logs with a and b combinations of adding, subtracting, division and multiplication. It took me a long time, but I was able to find the expression 3rd Log = ab /(a-b). Later in part six, I was able to confirm that the expression I found was the general expression I was looking for the whole time. After investigating further into the expression, I came to a conclusion that Log b/a X = cd/(c-d) is true. There is limitation to this expression, however. If a, b, or x ≠ (0, 1)U(1, ∞).
As I finished investigating the bases of common logarithms, I came to a conclusion that you can get third logarithm’s answer from the two Logs if I use expression Log b/a X = cd/(c-d) if a, b, and x = (0, 1)U(1, ∞), and x has to be constant in both Logs.
