L&T, 1..6
L&T, 1..1.2.4
Here we summarise the properties of the powers.
First of all the product of two powers,
(2.1) |
e.g., , and (we see that and do not have to be integers (whole numbers)). Question: Evaluate .
If we take the power of a power, we multiply the exponents,
(2.2) |
e.g., . This again works for not integers. Question: Evaluate .
If the exponent is we are taking the th root of ,
(2.3) |
e.g., , . If then . This can be shown by taking both sides to the power ,
The number is often taken to be an integer, but it does not have to be. (E.g., .)
If we take a number to the power zero, we find
(2.4) |
This follows from , and therefore . (Note that there is a slight problem with : for . One usually defines .)
If we take a number to a negative power, we write the result as a fraction involving a positive power,
(2.5) |
since . Therefore . E.g., .
Remember that
(2.6) |
and not
(2.7) |
As an example, , but .
L&T, 1..6.2
The exponential function is a special case of a power, where , with (Euler’s number). One also writes instead of .
As we can see from Fig. 2.1 , is never less than for any . From the properties of powers we know that . This function is also shown in Fig. 2.1 , and is positive as well.
Differential (derivative w.r.t. ) of is , i.e.,
(This is the only function with the property that the derivative equals the function itself.)
If then (this is a form of the chain rule, which will be discussed later), e.g., if then .
Example 2.1:
Discuss exponential growth/decay.
Solution:
Exponential growth or decay is ruled by the form . For we have decay, for we have growth. From the derivative, we see that this arrises when the change in is proportional to the number present. Examples are population growth, radioactive deay, ….
L&T, 1..6.3
L&T, 1..6.3.1
The inverse of a function is defined such that if , then .
A graph of the logarithm is shown in Fig. 2.2 . If we swap the and axes, we recognise the exponential. Normally we use logs to base (inverse of )- called natural logarithms, hence the name , but we also write
L&T, 1..6.3.2
Just as for the logarithm corresponding to base (i.e., the inverse of ) for other bases we have . Here we use the notation that if we mean to base, say, 10 we write , i.e., if , .
It may help you to remember that a logarithm tries to extract a power from a number, e.g. the extract the power of from a number.
Using this we can change from one base to another. Let , then . Now let (or ), so . Therefore , so , . Hence . Question: Determine such that .
If then
(Remember that the differential of is , not the integral! This is a common error!)
Using the fact that , i.e., the product of exponents is the exponent of the sum, we conclude that the inverse relation holds for logarithms. Thus, the logarithm of a product is the sum of the logarithms,
Example 2.2:
The magnitude of a start is defined as . Explain how changes if increase by one unit.
Solution:
The new intensity satisfies . Using the properties of the logarithms, we find that
Example 2.3:
An unresolved doube-star has magnitude 7. Find the individual magnitudes, assuming that both stars have the same one.
Solution:
Since intensities add up, we have . Thus we conclude that .