# Logarithm Bases Portfolio

by zostale (student)

Logarithm Bases

8/12/2012

IB-1 Mathematics

Introduction:

This portfolio is based on the mathematical concept of Logarithm Bases. The aim is to solve logarithm and exponential equations using log laws and base theorems and also to use and apply other mathematical concepts like series and sequences and exponents and relate them to logarithm concept. The portfolio is subdivided into three parts. Each part has a different question.

Sequence 1 :

->General: 2^n
Explanation for the next two terms in the sequence: Terms # 6 and 7 are the succeeding terms in this sequence. The argument being the same (8) in all the first five terms of the sequence there is a pattern observed in the bases which is the base of the first term is 2
1 , the second term 22 and hence according to the exponential pattern the base of the sixth and seventh term has to be 26 and 27 which is 64 and 128.

To find the nth term of this sequence in form of p/q:                                                                                    Log 2 8, Log 4 8, Log 8 8, Log 16 8, Log 32 8, Log 64 8, Log 128 8

The bases can be written as 21, 22, 23 up to 27. Hence the argument being same(8) bases can be generally denoted by 2n where n is the number of term.

Hence the logarithm equation for the nth term is : Log 2^n 8

But we have to express in fraction form(p/q form hence)

Let Log 2^n 8 = y,

Then according to logarithm law which states that if logab= c then ac=b

2ny= 8  2ny= 23

ny= 3(according to exponents law) and y= 3/n

Therefore the expression is: Log 2^n 8 = 3 / n

Sequence 2:

Explanation for the next two terms: # 5 and 6 terms are continuation of the series and they follow a pattern. The Argument (A) remains the same throughout the series – 81. The pattern with the bases was 3^n hence the 5 Th and 6 Th term bases being 35 and 36 which is 243 and 729 which were put as bases for the respective terms.

To find the nth term of this sequence in form of p/q:                         ...