Logarithms. In this investigation, the use of the properties of logarithms will be used to identify patterns, relationships, and limits of logarithms and sequences.

Introduction

In this investigation, the use of the properties of logarithms will be used to identify patterns, relationships, and limits of logarithms and sequences. A logarithm is an exponent in which the base must have in order to receive the number. All values found will be in the form p/q to further show a relationship with the terms.

Part 1

In the first sequence given, when examined closely, you will see that the log number remains the same throughout the part of the sequence given. You will also notice that it is the base of the logarithms that is altered.

log28, log48, log88, log168, log328, …

To begin, it would prove to be useful to figure out what sort of sequence it really is. By looking that the logarithms, it is safe to conclude that the sequence given cannot be an arithmetic sequence; the reason being that the bases of each logarithm are different, therefore, there is no common difference that describes the sequence as arithmetic.

Since it has been proven that the sequence cannot be arithmetic, the only type of sequence left, is a geometric sequence. A geometric sequence is a sequence whose consecutive terms are multiplied by a fixed, non-zero real number called a common ratio. Therefore, to prove that the sequence is indeed, geometric, the common ratio must be found.

When the sequence is examined closely, you will notice that the log number remains the same throughout the sequence and that it is the base of the logarithm that changes with every term. Understanding this, you can use the bases to find the common ratio of the sequence by dividing the consecutive terms by the terms preceding them.

After doing this, it is found that the common ratio of the sequence is 2. The process of evaluating the expression thus can begin…

log28, log48, log88, log168, log328, …

4/2 = 2 , 8/4 = 2 , 16/8 = 2 , 32/16 = 2

One way of evaluating what the values of the next two terms are is by using the formula logmnmk where m is the base and number (constant number), k is the exponent in which m is raised, and n is the term number

log28 = log21(23) , log48 = log22(23) , log168 = log24(23) , log328 = log25(23)

After finding this, you can use the Change-of-Base formula to convert the logarithm into a ratio between the two components of the logarithm.

log21(23) = log23 / log21

log22(23) = log23 / log22

log24(23) = log23 / log24

log25(23) = log23 / log25

Then, according to the power property of logarithms, the exponent of a logarithm can be moved to the front of the log as a multiplier which can be further simplified by canceling out log2 on each side of the fraction. This gives the values in the form p/q.

log23 / log21 = 3log2 / 1log2 = 3/1

log23 / log22 = 3log2 / 2log2 = 3/2

log23 / log23= 3log2 / 3log2 = 3/3 = 1

log23 / log24= 3log2 / 4log2 = 3/4

log23 / log25= 3log2 / 5log2 = 3/5

After evaluating the logarithms, you can see that the denominator of each value is the same as the term number. Therefore the conclusion that the value of the sequence can be calculated by using the expression 3/n where n is the term number. Using this expression, the value of the 6th and 7th terms may be found.

Value of 6th term Value of 7th term

3/6 3/7

Multiply the ratios by log2 and rewrite the logarithm using Change-the-Base.

3log2 / ...