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 any term in a sequence is found. This basic formula is also used to find the following terms that haven’t been found yet. Any term in a sequence is defined as the nth term, because “n” can represent any number. The expression used was in the form of p/q, where p,q εΖ. “p” is equal to lnb, and “q” is equal to lna from the expression logab=ln a/ln b, where ln stands for natural logarithm.
The system of natural logarithms uses the number called "e" as its base. e is the base used in natural logarithms in calculus. It is called the "natural" base because of certain technical considerations (lne=1)
ex has the simplest derivative. e can be calculated from the following series involving
factorials:
e is an irrational number, whose decimal value is approximately
2.71828182845904.
To indicate the natural logarithm of a number, we use the notation "ln."
ln x means logex (taken from: ).
After understanding natural logarithms, it makes it easier to understand the expression
used when dealing with the sequences mentioned above. The expression used to find the nth term (tn) of the first sequence is tn= log2n 8 = ln 8/ ln 2n. The expression used to find the nth term (tn) of the second sequence is tn= log3n 81= ln 81/ ln 3n.The expression used to find the nth term (tn) of the third sequence is tn= log5n 25 = ln 25/ln 5n. The expression used to find the nth term (tn) of the fourth sequence is tn= logmn mk = ln mk/ln mn. The answers can be easily justified by using the highly technological technology in a Texas Instruments graphing calculator. Considering log2 8=3, when plugging in ln 8/ ln2, the calculator provides the answer of 3, which is equivalent to the correct answer mentioned before.
Logarithms in Sequences: Calculating Logarithms
As mentioned earlier, logarithms can be solved and produce number answers, such as log28=3. Using the expressions from above this logarithm can also be solved like this: ln 8/ ln 2=3. Given this, consider the following logarithmic sequences:
-
log4 64, log8 64, log32 64
-
log7 49, log49 49, log343 49
-
log1/5 125, log1/125 125, log1/625 125
-
log8 512, log2 512, log16 512
Each of these logarithms can be calculated by using the formula logab=ln a/ ln b. In row 1, the calculated answers are as follows: ln 64/ ln 4, ln 64/ ln 8, ln 64/ ln 32. In row 2, the calculated answers are as follows: ln 49/ ln 7, ln 49/ ln 49, ln 49/ ln 343. In row 3 the calculated answers are as follows: ln 125/ ln (1/5), ln 125/ ln (1/125), ln 125/ ln (1/625). In row 4 the calculated answers are as follows: ln 512/ ln 8, ln 512/ ln 2, ln 512/ ln 16.
The answer obtained in the third column of each row can be deduced by just that of the first and second. If you didn’t notice already, a pattern exists. The answer of the third row can be obtained by keeping the same numerator while multiplying the coefficients of the first and second row’s denominators to produce the coefficient of the third row’s denominator. So basically, the base of the first log is multiplied by the base of the second logarithm to form the base of the third logarithm, while keeping this in mind the coefficient of the third logarithm stays the same as the other two logarithms.
Ex: ln 64/ ln 4, ln 64/ ln 8. ln 64 remains in the third logarithm as its numerator; and ln 4 is multiplied by ln 8 = ln (4 * 8) = ln 32. So ln 64/ ln 32.
A few more sequences can be formed like the previous ones:
-
log9 81= ln 81/ ln 9, log81 81= ln 81/ ln 81, log729 81= ln 81/ ln 729
-
log6 6= ln 6/ ln 6, log36 6= ln 6/ ln 36, log216 6= ln6/ ln 216
These sequences also follow the same patterns. In the first sequence, 9 from the first term is multiplied by 81 in the second term to form 729 in the third term; and in the second sequence, 6 from the first term is multiplied by 36 from the second term to form 216 in the third term.
Logarithms in Sequences: Finding the General Statement/ Concluding Statements
Consider the following: let loga x= c and logb x= d. given this you can find the general statement that expresses logab x, in terms of c and d, which is cd/c+d. This can only be tested using a calculator, so from the first sequence plug in ln 64/ ln 4, ln 64/ ln 8, and ln 64/ ln 32 all separately. Consecutively you should get 3, 2, and 1.2. Now convert 1.2 into the fraction of 6/5. From this you can notice that 6 is made when multiplying 3 and 2, and 5 is made when adding 3 and 2. Now try another sequence. From sequence 3, plug in ln 125/ ln (1/5), ln 125/ ln (1/125), and ln 125/ ln (1/625) all separately. You should get -3, -1, and -.75 respectively. Now lets try the formula of cd/ c+d. when plugged in you should get, (-3 * -1)/ (-3 + -1), which is reduced to 3/ -4, which is equal to -.75. This proves the statement to be valid. With formulas comes a limitation. However it seems as though this one lacks a limit in terms of a, b and x. It seems this formula of cd/c+d is only useful when trying to find the calculation of a logarithm using its roots. By splitting up a logarithm into two smaller ones, you can find its solution easier by using the formula. This can be helpful when used in certain situations; however it can be useless when unneeded in smaller logarithms. In terms of a, b and x, x must repeat in order to use this expression. To find the general statement of cd/ c+d, was by reverse solution. I found the answer of the third term and tried to find out how this answer could be derived by using the values calculated from the first two terms.
Word count: 1475
