# IB Pre-Calculus Logarithm Bases General Information: Logarithms A logarithm is an exponent and can be described as the exponent needed to produce a certain number.

Elliott

International Baccalaureate

IB Pre-Calculus

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Logarithm Bases

January 6, 2009

Student Number: 1208769

C. Leon King High School

Andre Elliott

January 6, 2009

Dr. Stone

IB Pre-Calculus

Logarithm Bases

General Information: Logarithms

A logarithm is an exponent and can be described as the exponent needed to produce a certain number.

For Example: 23=8, from this you would say that 3 is the logarithm of 8 with base of 2 (log28). 2 is written as a subscript, and 3 is the exponent to which 2 must be raised to produce 8.

The formula or definition that is used in logarithms is: logbx=e, so be=x. So base(b) with exponent(e) produces x. So from the example, 2 is the base(b), 8 is x (the number produced), and the exponent(e) is 3. So, 23=8.

Logarithms in Sequences: Introduction

Since logarithms can be solved (log28=3) to form numbers, this means logarithms are just another way to represent a number; and since numbers can be in sequences, so can logarithms. Given this, consider the following sequences:

1. log2 8, log4 8, log8 8, log16 8, log32 8, …
2. log3 81, log9 81, log27 81, log81 81, …
3. log5 25, log25 25, log125 25, log625 25, …
4. logm mk, logm2 mk, logm3 mk, logm4 mk

Sequence 4 is the general statement that is used to reflect the previous sequences. So, consider the first sequence; it starts out with a base(m) of 2, the second term has a base(m) of 4, which is equal to 22, then the third term has a base(m) of 8, which is equal to 23, etc. This can be broken down into a basic sequence just contain the bases: 2, 4, 8, 16, 32, …; this can also be written as 2, 22, 23, 24, 25,…; this idea can be applied to each of the sequences represented above.

Given this, the next two or more terms of each sequence can be found. In the first sequence, the next three terms are: log64 8, log128 8, and log2568. In the second sequence the next four terms are: log243 81, log729 81, log2187 81, and log6561 81. In the third sequence, the next four terms are: log3125 25, log15625 25, log78125 25, and log390625 25. In the fourth sequence, the next four terms are: logm5 mk, logm6 mk, logm7 mk, and logm8 mk.

Logarithms in Sequences: Forming Expressions/ Natural Logarithms

In logarithmic sequences, expressions can be found in order to justify how ...