# IB SL Math Portfolio- Logarithm Bases

IB Math SL Portfolio Type 1

General Introduction:

In its simplest terms, a logarithm is an exponent. It is an exponent needed to produce a given number from a specific base. It is written in the form loggh=j, which denotes gh=j (g to the power of h equals j).

LOGARITHM BASES

Consider the following sequences. Write the next two terms of each sequence.

log28, log48 , log88, log168, log328, ... log648, log1288, ...

log381, log981, log2781, log8181, … log24381, log72981, ...

log525, log2525, log12525, log62525, … log3,12525, log15,62525, ...

:

:

:

logmmk, logm^2mk, logm^3mk, logm^4mk, … logm^5mk, logm^6mk , …

Find an expression for the nth term of each sequence. Write your expressions in the form p/q, where p and q are integers. Justify your answers using technology.

The first sequence can be otherwise written:

log2^123, log2^223 , log2^323, log2^423, log2^523, ...

Here, I noticed a pattern: the base of each logarithm is 2n. Using this knowledge and the concepts of the change of base rule and verifying my theories with a GDC calculator, I developed the expression shown here by evaluating each term.

(Each term to be evaluated can be checked by using a GDC calculator. For example, log28 reads”the exponent of 2 that yields 8”. As these are relatively small numbers, mental math can be used to evaluate the statement, which simplifies to 3. To verify this response, one could type 2^3 in their calculator and press “enter” or “solve”. One could also use the “solver” function on a GDC calculator to check their math by simply pressing the “math” key, then pressing “solve”. The original statement can be written 2x=8, but in order to input this into the solver function, it needs to be set to zero. So, one could input 2x -- 8=0 and press solve to check their answer).

The first sequence looked like this:

log2^123 = 3/1

log2^223 = 3/2

log2^323 = 3/3

log2^423 = 3/4

log2^523 = 3/5

:

:

log2^n23 = 3/n

The nth term of this sequence can be expressed as 3/n.

I completed the same steps to discover the general expressions of each of the following sequences.

log381, log981, log2781, log8181, ... can be written log3^134, log3^234, log3^334, log3^434, …

log3^134 = 4/1

log3^234 = 4/2

log3^334 = 4/3

log3^434 = 4/4

:

:

log3^n34 = 4/n

The nth term of the second sequence can be expressed as ...