Use the = rule and then we can change the base. We change the base to 10.
Then we apply the rule: = a
We can cross away on both sides.
What remains is: . Derived from this we can conclude that the general expression for the nth term of each sequence in the form thus is .
Examples to justify this statement using technology:
= ()
= = 1,5
=
=
-
Now calculate the following, giving your answer in the form , where p, q .
The answer was . This form will be used.
= ()
= = 3
= ()
= = 2
= ()
= = 1,2
2. = ()
= = 2
= ()
= = 1
= ()
= ≈ 0,67
3. = ()
= = -3
= ()
= = -1
= ()
= = -0,75
4. = ()
= = 3
= ()
= = 9
= ()
= = 2,25
- Describe how to obtain the third answer in each row from the first two answers. Create two more examples that fit the pattern above.
1.
As we can see, n=2 in the first logarithm and in the second logarithm n=3. If we add these together, we get n=2+3=5. That means that in the first row, the third answer is obtained by adding the first two n up together. The pattern is therefore that you add up the two n in front of the next logarithm.
The next two examples which would fit in the pattern would therefore be:
, =
2. =
As we can see, n=1 in the first logarithm and in the second logarithm n=2. There is an arithmetic increase, with the fixed number of 1. The next number in the second row will therefore be n=3. The pattern thus is that there is an arithmetic increase with the fixed number of 1.
The next two examples which would fit the pattern would therefore be:
, =
3. =
As we can see, n=-1 in the first logarithm and in the second logarithm n=-3. If we add these up together, we get n= -1 + -3 = -4. That means that the third row, the third answer is obtained by adding the first two n up together. The pattern is therefore that you add up the two n in front of the next logarithm.
The next two examples which would fit the pattern would therefore be:
, =
4. =
As we can see, n=3 in the first logarithm and in the second logarithm n=1. Something has to have been before the n=3, which means in front of the first logarithm. It should have started from n=0, as we can derive, from the second to the third logarithm, wherein there is an increase in n of 3. The pattern is therefore that you add 3 and you subtract 2 from the next logarithm and so forth.
The next two examples which would fit the pattern would therefore be:
, =
-
Let =c and =d. Find the general statement that expresses in terms of c and d.
=c and =d then find
One law of logarithms state that:
We use the change of base rule:
=c then =x =d then =x
= =
We are taking logarithms in base x:
= =
d =
Derived from we can state that:
+ =
If we change the base again we get the following equation:
We substitute :
The following step is to multiply both sides by cd:
The general statement that expressesin terms of c and d thus is:
- Test the validity of your general statement using other values of a, b, and x.
=c and =d
1. Example: a=2, b=4, x=8
Check with the general statement:
c = = = = 3
d = = = = 1,5
= = 1
General statement justified.
2. Example: a=5, b=125, x=25
Check with the general statement:
c = = = = 2
d = = = ≈ 0,67
= = 0,5
General statement justified.
3. Example: a=10000, b= , x=10
≈ 0,33
Check with the general statement:
c = = = = 0,25
d = = = ≈ -1
= = ≈ 0,33
General statement justified.
- Discuss the scope and/or limitations of a, b, and x.
The limitations of logarithms are usually, as stated in the second sentence of this internal assessment:
a>0, a1, b>0;
which would mean for this question that the limitations are:
a>0, b>0, a≠1, b≠1, x>0
We can do a check for this:
Example: a=-2, b=2, x=4
It is impossible to power a function which results in a negative number, in this case a = -2. With these numbers: , n>0.
The same applies for b, that a>0.
Example 2: a=10, b=1, x=100
N=0 as , which makes it impossible, as you have to divide that number and = error/not possible.
The sample applies for a, that b≠1.
Example 3: a=4, b=8, c=-8
Same reason as in example 1: it is impossible to power a function which results in a negative number, in this case x (=-8). With these numbers: , k>0.
As a>0 and b>0, the product x should always be greater than 0, therefore x>0.
To sum up again:
a>0, b>0, a≠1, b≠1, x>0
- Explain how you arrived at your general statement.
One law of logarithms state that:
We use the change of base rule
=c then =x =d then =x
= =
Take logarithms in base x:
= =
d =
Derived from we can state that:
+ =
If we change the base again we get the following equation:
We substitute :
The last and following step is to multiply both sides by cd: