The property that must be used in all the proofs is: logb(b^y) =y
<h3>What are the proofs or rules of logarithms?</h3>
The proofs are statements that are used to validate or invalidate a logarithmic expression
There are several proofs of logarithms; some of them are:
- Product rule
- Quotient rule
- Power rule
- Change of base
The common property in the first three proofs (listed above) is:
This is so because, it links all the three proofs
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Answer:
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Explanation:
Answer:
F '( 1 ) = 84
Explanation:
We will differentiate
F
and wade our way slowly and methodically through finding the derivative of the right. First with the outermost function and chain rule:
F
(
x
)
=
f
(
x
f
(
x
f
(
x
)
)
)
F
'
(
x
)
=
f
'
(
x
f
(
x
f
(
x
)
)
)
(
d
d
x
x
f
(
x
f
(
x
)
)
)
Now applying the product rule:
F
'
(
x
)
=
f
'
(
x
f
(
x
f
(
x
)
)
)
[
f
(
x
f
(
x
)
)
+
x
(
d
d
x
f
(
x
f
(
x
)
)
)
]
Reapplying the chain rule:
F
'
(
x
)
=
f
'
(
x
f
(
x
f
(
x
)
)
)
[
f
(
x
f
(
x
)
)
+
x
(
f
'
(
x
f
(
x
)
)
+
(
d
d
x
x
f
(
x
)
)
)
]
F
'
(
x
)
=
f
'
(
x
f
(
x
f
(
x
)
)
)
[
f
(
x
f
(
x
)
)
+
x
f
'
(
x
f
(
x
)
)
+
x
(
d
d
x
x
f
(
x
)
)
]
Product rule once more:
F
'
(
x
)
=
f
'
(
x
f
(
x
f
(
x
)
)
)
[
f
(
x
f
(
x
)
)
+
x
f
'
(
x
f
(
x
)
)
+
x
(
f
(
x
)
+
x
f
'
(
x
)
)
]
We could simplify this a little more, but I'm dubious it would help. Evaluating at
x
=
1
:
F
'
(
1
)
=
f
'
(
f
(
f
(
1
)
)
)
[
f
(
f
(
1
)
)
+
f
'
(
f
(
1
)
)
+
1
(
f
(
1
)
+
f
'
(
1
)
)
]
F
'
(
1
)
=
f
'
(
f
(
2
)
)
[
f
(
2
)
+
f
'
(
2
)
+
2
+
4
]
F
'
(
1
)
=
f
'
(
3
)
(
3
+
5
+
2
+
4
)
F
'
(
1
)
=
6
(
14
)
F
'
(
1
)
=
84
Why dont you try D if im wrong you can blame me for failing the test
The statement that explains such a relationship is that as the percent of woodland increases, the number of deer observed in a group decreases quickly at first and then more slowly
- Nonlinear relationship is a type of relationship that changes that takes place in the output change not in direct proportion to alterations (changes) in any of the inputs.
- This type of relationship does not produce any straight line but produces a curve line.
- Negative relationship is simply known as negative or inverse correlation that occurs between two variables. That is when that one variable increases, the other decreases, and vice-versa. As in the case of the woodland and the beer. When woodland increases, the deer decreases and vice versa
Conclusively, we can say that as the statement that explains such a relationship is that as the percent of woodland increases, the number of deer observed in a group decreases quickly at first and then more slowly
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