I created the last two terms in this sequence, terms 6 and 7 (5 & 6), simply by doubling the base of the logarithm for each term.
By following these sequences a pattern can be shown. The base of each term in the sequences changes but the index is constant.
Finding an expression for the nth term for each sequence:
To find the nth term we need to solve the logarithms by using the change of base law.
-
log8 = = 3 (using GDC)
log8 = = 1.5 or
log8 == 1 or
log8 = = .75 or
log8 = = .6 or
log8 = = .5 or
Thus, the nth term is where p is 3 and q is n.
Justify using technology:
In this manner, I created a formula to find the numerical equivalence for the nth term of the sequence in the form, where both p and q are integers.
-
log81 = 4 log81 = 1 or
log81 = 2 or log81 = 0.8 or
log81 = 1.667 or log81 = 2/3 or
Thus the nth term is where p is 4 and q is n.
-
log25 = 2 log25 = 0.5
log25 = 1 log25 = 0.4
log25 = 2/3 log25 = 1/3
Thus the nth term is where p is 2 and q is n.
-
logm = k logm= k/4
logm = k/2 logm=k/5
logm = k/3 log m= k/6
Thus the nth term is where p is k and q is n.
Describing how we obtain log(64) from log(64) and log(64).
In order to obtain the answer of the third logarithm, the product of the first and second logarithms must be divided by the sum of the answers of the first and second logarithms.
log(x), log(x) log(x)
We use the formula:
logb =
log(x) = log4 = [1]
log(x) = log8 = [2]
Add [1] and [2]:
log4 + log8 = +
And we have:
log32 =
Therefore the formula we obtain:
log(x) = or log(x) =
By replacing the value of x = 64 we get,
log(64) =
log(64) = 3 & log(64) = 2 (using GDC)
Thus, log(64) =
Describing how we obtain log(49) from log(49) and log(49).
Thus by using the above formula above:
log(x) =
log(49) = log(49) log(49)
log(49) + log(49)
Solving the equations,
log(49) = 1
log(49) = 2
Therefore, log(49) =
Describing how we obtain log (125) from log (125) and log (125).
Thus, by using the formula
log(x) = ,
log (125) =
+
Solving the equations
log (125) & log (125) by change of base log
= -3 = -1
Therefore, log (125) = -
Describing how we obtain log(512) from log(512) & log(512).
Thus, by using the formula
log(x) =
We obtain,
log(512) =
Solving the equations,
log(512) = 3 , log(512) = 9
Therefore,
log(512) =
= 2.25
Example 1:
1st term - log
2nd term- log
3rd term- y
Therefore,
y =
y =
y = log
y =
Example 2:
1st term- log 0.01
2nd term- log 0.01
3rd term- y
Therefore,
y =
y =
y = log0.01
y = -0.8692
Expressing, log(x) in terms of c and d
Let log(x) = c and log(x) = d
Formula:
log(M) = log(x) +log(x) = log(ab)
+=
Substituting c and d
+ =
= log(x)
OR
log(x) = c = c
log (x) = d = d
Therefore,
log(x) =
= +
= log(x)
Thus, the general statement is.
To test the validity of the statement we need to by putting different values of a, b and x in the equation.
Scope/Limitations
After testing the validity of the general statement by putting numerous values of a, b and x we can say that some values are still undefined.
The base of the log must be positive and not equal to one so we can say: a > 0, b > 0, a1, b1, they both can’t be negative because each of them is shown separately in the logarithm. Also ab >1 and a.
Finally, the argument (x) must be positive, so:
x > 0
General Statement
The general statement was arrived by two ways:
1st way:
By using change of base log formula log(M) = .
We simplified log(x) and used log10ab = log10 a + log 10 b in the denominator.
Now by simplifying loga x= c and logb x = d by changing the base logs by the formula above and making loga x and logb x as the subject of the formula and substituting in the first part we get the general statement as .
2nd way:
We simplified the statement logab x by making the bases similar.
Now in the equations loga x= c and logb x = d we used the formula to make the bases similar i.e. and making loga x and logb x as the subject of the formula.
Then we substituted it in the simplified version of logab x . By substituting the values we get the general statement.
Through this general statement we can find out the value of the third term in a sequence when the first two terms are given.
Conclusion
In conclusion, after applying the given knowledge of logarithms, a general statement was formed to prove the validity of various sequences. Also through the use of technology, general statements were proven and theories were confirmed.
