Recall that when we have a function y = f (x), the inverse function x = f−1(y) may exist, and we saw in Section 2.9 that it can be found nicely in some cases (as in the temperature example). What is the inverse function of the natural exponential function, y= f (x) = ex? As yet, we have no device to express x in terms of y in this case, so we shall introduce a new notion as follows:
x= logey. (3.36)
This function is called the logarithmic function with base e (remember, we were looking at the exponential function with base e), or we just call it the natural logarithmic function. It is an inverse function of y= ex.
Most of the time, we write x = logeyin an alternative way:
x= ln y. (3.37)
In fact, we can also define logarithmic functions with other bases as follows.
Definition 3.2 The logarithmic function with base α, where α > 0 and α= 1, is denoted by logα and is defined by:
x= logαyif and only if y= αx. Alternatively we can write:
x= logαy⇔ y = αx. (3.38)
Let us do some simple exercises.
Question Obtain x.
(1) x= log216.
(2) x= log101.
(3) x= ln e2. (4) x= ln 1.
Solution (1)
x= log216 2x = 16
x= 4.
(2)
x = log101 10x = 1
x = 0.
(3)
x= ln e2 x= logee2 ex = e2
x= 2.
(4)
x= ln 1 x= loge1 ex = 1
x= 0.
Exercise 3.12 Exponentials and logarithms.
77 3.6 Logarithms: how many years will it take for my money to double?
Two bases are commonly used for the logarithm. One is the one we have discussed, e, the natural logarithm (ln). The other is 10. We call log10 the common logarithm (since we use a decimal system of numbers, it is likely to be the most common value for the base). Most scientific calculators will have the facilities of log10and ln but, as discussed in Preface, it is irrelevant in the spirit of the book. When you end up with expressions including e and/or ln, leave it as long as further simplification is not possible.
3.6.1 Logarithm properties
The logarithmic function has many important properties. I will list those, and they will be assumed knowledge for the rest of the book. I will show in the following exercise that the first property must hold, but will leave the rest for you to show, since it can be done in a similar fashion. Some exercises are provided after the list of properties.
Logarithm properties Property 1
logax+ logay= loga(xy). (3.39) Property 2
logax− logay = loga
x y
. (3.40)
Property 3
logaxb= blogax. (3.41)
Property 4
logax= logab
· logbx
. (3.42)
Property 5
logax= 1
logxa. (3.43)
Question Show that Equation (3.39) holds.
Solution Let logax= m and logay= n. So the LHS of Equation (3.39) is m + n. Note also that am= x and an= y by the definition of the logarithm, and hence xy = aman= am+n.
In the meantime, the RHS of Equation (3.39) is: loga(xy)= loga(am+n). By the definition of the logarithm, it is equal to m+ n.
Hence we have shown that the LHS and the RHS of Equation (3.39) are equal.
Exercise 3.13 Deducing Equation (3.39).
Here are some exercises for you to get used to logarithms.
Question Express x in terms of a and b, provided that log102= a and log103= b.
(1) x= log102000.
(2) x= log10
1 9. Solution (1)
x= log102000
= log102+ log10100
= a + 2.
(2)
x= log10
1
= log1019− log109
= 0 − log1032
= −2log103
= −2b.
Exercise 3.14 Logarithm calculation.
In passing, the graph of the natural logarithmic function y = ln x is depicted in Figure 3.6. Again, at this stage, it suffices to convince yourself that the graph is shaped as shown in the figure by plotting several points. We will learn how to properly sketch the logarithmic function in Chapter 5. The horizontal intercept is (1,0) because ln 0= 1. The value of the function y increases as x increases but at a decreasing rate. If you imagine that there is a mirror on the line y = x, the graph of the natural logarithmic function y = ln x is a mirror image of its inverse function y= ex.
3.6.2 How many years will it take for my money to double?
Now that we know how to deal with the logarithms, we are able to tell how many years it will take for our money to double, given the information about compounding. Let us think how many years it will take for our money to double at the effective rate of 5 per cent.
Recall first that the effective rate is the equivalent rate compounded annually. If we denote the principal by P , then the compound amount over t years can be written as P(1+ 0.05)t. This has to be equal to twice as much as P , which is 2P , in which case:
2P = P (1 + 0.05)t. (3.44)
79 3.6 Logarithms: how many years will it take for my money to double?
1
1
y y = ex
y = x y = ln x
x 0
Figure 3.6 A graph of the natural logarithmic function.
Rearranging this equation yields:
1.05t = 2. (3.45)
Taking (the natural) logarithms of the both sides of this equation yields:
ln 1.05t = ln 2. (3.46)
Now, using one of the logarithm properties, we rearrange the left hand side of this equation in order to solve for t:
tln 1.05= ln 2. (3.47)
Hence:
t = ln 2
ln 1.05(years). (3.48)
This is the answer and you can stop here. If you have a scientific calculator, you can check that the value of t in (3.48) is roughly 14.21. Remember this value in reference to the next exercise.
Question How many years will it take for our money to double if interest is compounded quarterly at a nominal rate of 5 per cent?
Solution
2P = P
1+0.05 4
4t
ln 2= 4t ln 1.0125 t = ln 2
4 ln 1.0125
≈ 13.95.
The answer shows us that it will take (approximately) 13.95 years for our money to double if interest is compounded quarterly at a nominal rate of 5 per cent. Recall that if interest is compounded annually at 5 per cent (the example we saw above) it takes (approximately) 14.21 years for our money to double.
This finding is consistent with our discussion on the compound interest in the previous section. Given a positive nominal rate, recall that, the more frequent the compounding, the greater is the compound interest. It follows that, given a positive nominal rate, the more frequent the compounding, the less time it will take for our money to double. Hence, in this question, it does not take as long as 14.21 years to double our money.
Exercise 3.15 Doubling funds.