IB Mathematics SL Portfolio: Logarithmic Bases
IB Mathemetics SL Portfolio: Logarithmic Bases
In this portfolio task, I will investigate the rules of logarithms by identifying the logarithmic sequences. After identifying the pattern, I will produce a general statement which defines the sequence. I will then test the validity of my general statement by using other values. I will finally conclude the portfolio task by explaining how I arrived to my general statement and its limitations.
Consider the following sequences. Write down the next two terms of each sequence.
For the first row, the next two terms of the sequence would be log64 8 and log128 8. For the second row, the next two terms of the sequence would be log243 81 and log729 81. For the third row, the next two terms of the sequence would be log3125 25 and log15625 25.For the fourth row, the next two terms of the sequence would be logm5 mk and logm6 mk.
Find an expression for the nth term of each sequence. Write your expressions in the form p/q where p, q Z. Justify your answer using technology.
To find the nth term, I first solved the sequence on the first row. This is shown below.
Log2 8 = x Log4 8 = x Log8 8 = x Log16 8 = x Log32 8 =x
⇒2x = 8 ⇒4x = 8 ⇒8x = 8 ⇒16x = 8 ⇒32x = 8
⇒2x = 23 ⇒(22)x = 23 ⇒x = 1 ⇒(24)x = 23 ⇒(25)x = 23
⇒x = 3 ⇒2x = 3 ⇒4x = 3 ⇒5x = 3 ⇒x = 3/2 ⇒x = 3/4 ⇒ x= 3/5
Log64 8 = x Log128 8 = x
⇒64x = 8 ⇒128x = 8
⇒(26)x = 23 ⇒(27)x = 23
⇒6x = 3 ⇒7x = 3
⇒x = 3/6 ⇒ x = 3/7
By looking at these answers for the first sequence, I noticed the fact that the denominator of each answer shows the nth term. For example, my answer to Log2 8 (which is the first term) was 3 (or 3/1) and my answer to Log4 8 (which is the second term of the sequence) was 3/2. By looking at my answers to the first sequence, I found out that the nth term could be found by the expression 3/n.
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In this portfolio task, I will investigate the rules of logarithms by identifying the logarithmic sequences. After identifying the pattern, I will produce a general statement which defines the sequence. I will then test the validity of my general statement by using other values. I will finally conclude the portfolio task by explaining how I arrived to my general statement and its limitations.
Consider the following sequences. Write down the next two terms of each sequence.
For the first row, the next two terms of the sequence would be log64 8 and log128 8. For the second row, the next two terms of the sequence would be log243 81 and log729 81. For the third row, the next two terms of the sequence would be log3125 25 and log15625 25.For the fourth row, the next two terms of the sequence would be logm5 mk and logm6 mk.
Find an expression for the nth term of each sequence. Write your expressions in the form p/q where p, q Z. Justify your answer using technology.
To find the nth term, I first solved the sequence on the first row. This is shown below.
Log2 8 = x Log4 8 = x Log8 8 = x Log16 8 = x Log32 8 =x
⇒2x = 8 ⇒4x = 8 ⇒8x = 8 ⇒16x = 8 ⇒32x = 8
⇒2x = 23 ⇒(22)x = 23 ⇒x = 1 ⇒(24)x = 23 ⇒(25)x = 23
⇒x = 3 ⇒2x = 3 ⇒4x = 3 ⇒5x = 3 ⇒x = 3/2 ⇒x = 3/4 ⇒ x= 3/5
Log64 8 = x Log128 8 = x
⇒64x = 8 ⇒128x = 8
⇒(26)x = 23 ⇒(27)x = 23
⇒6x = 3 ⇒7x = 3
⇒x = 3/6 ⇒ x = 3/7
By looking at these answers for the first sequence, I noticed the fact that the denominator of each answer shows the nth term. For example, my answer to Log2 8 (which is the first term) was 3 (or 3/1) and my answer to Log4 8 (which is the second term of the sequence) was 3/2. By looking at my answers to the first sequence, I found out that the nth term could be found by the expression 3/n.
For