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Indices and Logarithms
Number and Algebra : Module 31Years : 9-10
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- Knowledge of the index laws for positive integer powers.
- Facility with the arithmetic of integers and fractions.
- Facility with basic algebra.
- Familiarity with rounding numbers correct to a given number of decimal places.
Indices provide a compact algebraic notation for repeated multiplication. For example, is it much easier to write 35 than 3 × 3 × 3 × 3 × 3.
Once index notation is introduced the index laws arise naturally when simplifying numerical and algebraic expressions. Thus the simplificiation 25 × 23 = 28 quickly leads
to the rule am × an = am + n, for all positive integers m and n.
As often happens in mathematics, it is natural to ask questions such as:
- Can we give meaning to the zero index?
- Can we give meaning to a negative index?
- Can we give meaning to a rational or fractional index?
These questions will be considered in this module.
In many applications of mathematics, we can express numbers as powers of some given base. We can reverse this question and ask, for example, ‘What power of 2 gives 16? Our attention is then turned to the index itself. This leads to the notion of a logarithm, which is simply another name for an index.
Logarithms are used in many places:
- decibels, that are used to measure sound pressure, are defined using logarithms
- the Richter scale, that is used to measure earthquake intensity, is defined using logarithms
- the pH value in chemistry, that is used to define the level of acidity of a substance,
is also defined using the notion of a logarithm.
When two measured quantities appear to be related by an exponential function, the parameters of the function can be estimated using log plots. This is a very useful tool in experimental science.
Logarithms can be used to solve equations such as 2x = 3, for x.
In senior mathematics, competency in manipulating indices is essential, since they are used extensively in both differential and integral calculus. Thus, to differentiate or integrate a function such as , it is first necessary to convert it to index form.
The function in calculus that is a multiple of its own derivative is an exponential function. Such functions are used to model growth rates in biology, ecology and economics, as well as radioactive decay in nuclear physics.
We recall that a power is the product of a certain number of factors, all of which are the same. For example, 37 is a power, in which the number 3 is called the base and the number 7 is called the index or exponent.
In the module, Multiples, Factors and Powers, the following index laws were established for positive integerexponents. So positive integers and , and rational numbers and , we have:
- To multiply powers with the same base, add the indices.
aman = am+n.
- To divide powers with the same base, subtract the indices.
= am − n, (provided m > n.)
- To raise a power to a power, multiply the indices.
(am)n = amn.
- A power of a product is the product of the powers.
(ab)m = ambm.
- A power of a quotient is the quotient of the powers.
m= , (provided b ≠ 0.)
These laws also hold when a and b are real.
Show that ÷ = 6ab5.
We now seek to give meaning to other types of exponents. The basic principle we use throughout is to choose a meaning that is consistent with the index laws above.
The Zero Index
Clearly = 1. On the other hand, applying index law 2, ignoring the condition m > n,
we have = 50. If the index laws are to be applied in this situation, then we need to define 50 to be 1.
More generally, if a ≠ 0 then we define a0 = 1.
Note that 00 is not defined. It is sometimes called an indeterminant form.
(The explanation of this term is that one can find sequences of numbers of the form ab in which both a and b approach 0, but where the limit of the sequence is not 1 and indeed can be made to be any number we like, by a suitable choice of and For example, the terms of the sequence
1, 0, 0, 0,…
are all equal to 1, while the terms of the sequence
01, 0, 0, 0, …
are all equal to 0. In each case the form of the terms approaches 00.
A similar situation occurs with and so the expression is also often referred to as an indeterminant form.
(3a2b)0 = 1, assuming a and b are not zero.
The index laws also hold for the zero index.
If we examine the pattern formed when we take decreasing powers of 2, we see
24 = 16, 23 = 8, 22 = 2, 21 = 2, 20 = 1, 2−1 = ?, 2−2 = ?
At each step as we decrease the index, the number is halved. Thus it is sensible to define
2−1 = .
Furthermore, continuing the pattern, we define
2−2 = = , 2−3 = = , and so on.
These definitions are consistent with the index laws.
For example, = 23 − 4 = 2−1. But clearly, = .
Similarly, = 22 − 4 = 2−2. but clearly, = = .
We can confirm our intuition by considering and .
In general, for any non-zero number , and positive integer , we define
a−1 = and a−n = .
Note that all the earlier index laws also hold for negative indices.
We now extend our study of indices to include rational or fractional exponents. In particular, can we give meaning to 4?
Once again, we would like the established index laws to hold. Hence, squaring this expression we would like to say:
42 = 4 × 2 = 41 = 4.
Thus we define 4to be = 2.
In general we define a= for any positive number a.
Note that we have defined ato be the positive square root of We do this so that there is only one value for a.
Applying a similar argument, for consistency with the index laws, we define a = ,
a = , and so on.
In general, for any positive integer n and positive number a, define a = .
27 = 3, 16 = 2, (a6) = a4.
Assuming consistency with index law 3, we can write 82 = 8 × 2 = 8. But 8= = 2.
Thus, 8 = 4.
The notation 8means ‘the square of the cube root of 8’ which is equal to 4.
Note that we could also have expressed this as the ‘cube root of the square of 8’, which is, of course, also equal to 4, that is:
In general, if is a positive number and are positive integers, we define
a = ap or .
In words, we take the qth root of a and raise it to the power p.
Negative fractional indices
Finally, we can extend the indices to include negative rationals. For example,
8− = 8−1 = = .
a = .
We have now defined ax for any positive real number a and any rational number x.
It remains to check that the index laws also hold in this more general case. We will not go through the details. The following example outlines how this might be done in one particular case.
It is possible to give similar proofs that the other index laws also hold for negative integer and rational exponents.
Scientific notation, or standard form, is a convenient way to represent very large or very small numbers. It allows the numbers to be easily recorded and read.
The star Sirius is approximately 75 684 000 000 000 km from the sun. We can represent this number more compactly by moving the decimal point to just after the first non-zero digit and multiplying by an appropriate power of 10 to recover the original number. Thus
75 684 000 000 000 = 7.5684 × 1013.
If we move the decimal point 13 places to the right, inserting the necessary zeroes, we arrive back at the number we started with.
We can similarly deal with very small numbers using negative indices. For example, an Angstrom (Å) is a unit of length equal to 0.000 000 000 1 m, which is the approximate diameter of a small atom. We place the decimal point just after the first non-zero digit and multiply by the appropriate power of ten. Thus,
0.000 000 000 1 = 1 × 10−10. Hence, for example, the diameter of a uranium atom is 0.000 000 000 38 m which we may write as 3.8 × 10−10 m or 3.8 Å.
The index laws may be used to perform operations on numbers written in scientific notation.
Simplify (3.14 × 10−2)3 ÷ (7.1 × 10−8) giving your answer correct to one decimal place.
(3.14 × 10−2)3 ÷ (7.1 × 10−8) = (3.143 ÷ 7.1) × 102 ≈ 4.36044 × 102 ≈ 436.0 correct to 1 decimal place. In this case, we could leave this as the answer, or, if required, write is as 4.36 × 102.
Significant figures in scientific notation
Scientists and engineers routinely employ scientific notation to represent large and small numbers. Since all measurements are approximations anyway, they generally report the numbers rounded to a given number of significant figures. Thus, a number such as 2.1789 × 107 could be written as approximately 2.18 × 107. This is the same as rounding the number 21 789 000 to 21 800 000, that is, correct to three significant figures.
A given number may be expressed with different numbers of significant figures. For example, 3.1 has 2 significant figures, 3.14 has 3 significant figures and so on. To round a number to a required number of significant figures, first write the number in scientific notation and identify the last significant digit required. Then leave the digit alone if the next digit is 0, 1, 2 ,3 or 4 (in this case the original number is rounded down) and increase the last digit by one if the next digit is 5, 6, 7, 8 or 9 (in this case the original number is rounded up.)
Use a calculator to find correct to 4 significant figures.
We can use the calculator to find approximate values of for various rational values of 2x. We place these in a table and we can then plot the ordered pairs (x, 2x)to produce a graph of y = 2x.
Produce a table of values for the function y = 2x and use it to draw its graph.
A table of approximate values follows:
Note that although we have ‘joined the dots’ to form a smooth curve, we have not given any meaning at this stage to 2x when is an irrational number. We cannot deal with this problem at this stage.
We note the following features of the graph.
- the graph is increasing.
- the values increase quite rapidly as we move along the axis.
- on the left hand-side, the graph approaches, but never reaches, the axis.
Draw the graphs of y = 3x and y = 3−x on the set of axes.
An exponential equation is an equation in which the pronumeral appears as an index.
For example, 23x = 64 is an exponential equation.
We can see from the graph that the curve y = 23x and y = 64 the line only meet once,
so there is one unique solution to the exponential equation.
We can solve the equation as follows:
23x = 64
Hence 3x = 6, giving x = 2.
Solve 33 − x = 27x− 1.
How do we solve 2x = 7? The method used above does not work in quite the same way, since we do not know how to express 7 as a power of 2.
We will revisit this problem after we have looked at logarithms.
The exponential function is used to model growth − generally population growth in biology, but this may also include the growth of money via compound interest.
Suppose that a culture initially contains 1000 bacteria and that this number doubles
each hour. Thus, after
- one hour there are 1000 × 2 bacteria
- two hours there are 1000 × 2 × 2 = 1000 × 22 bacteria
- three hours there are 1000 × 22 × 2 = 1000 × 23 bacteria
and so on.
Following the pattern, if there are bacteria after hours then
N = 1000 × 2t.
This is an example of exponential growth.
Exponentials can also be used to model radioactive decay. Radioactivity is a natural phenomenon in which atoms of one element ‘decay’ to form atoms of another element by emitting a particle such as an alpha particle.
This is an example of exponential decay.
Exponential formulas have the form
P = A × Bt, where A, B are positive constants. If
- B > 1, we say that Pgrows exponentially,
- B < 1 we say that Pdecays exponentially.
For the rule y = 20 × 3t:
a Complete the table of values.
b Plot the graph of y against t.
c Find the value y, correct to 2 decimal places, when:
it = 0.5iit = 2.5 iiit = 2.8
It is easy to find values of x, such that 2x = 2 or 2x = 4, or 2x = 32. On the other hand, how do we solve the equation 2x = 10?
Problems such as this arise naturally when we deal with exponential growth and decay.
In the example above, we gave the formula for the mass of a radioactive substance to be M = 100 × tg.
If we ask the question, when is the mass equal to say 30g, then we need to solve t = 0.3 to find the time.
Just as taking a square root is the inverse process to squaring, taking a logarithm is the inverse process to taking a power.
Since 23 = 8, we say that log2 8 = 3. That is, the logarithm is the index in the equation
23 = 8. We read this as ‘the log of 8 to the base 2 is 3.’
To find the logarithm of a number a to the base b, we ask the question ‘what power do I raise b to, in order to obtain a?
So, to find for example, log3 243, we recall that 243 = 35, so log3 243 = 5.
Find log8 4.
One approach is to write, log8 4 = x and so 4 = 8x. Since both numbers are powers of 2, we can write 22 = (23)x = 23x.
Equating indices, 3x = 2, so x = .
Thus, log8 4 =.
(Indeed, 8= ()2 = 22 = 4.)
The relationship connecting logarithms and powers is:
x = logay means y = ax.
The number is called the base and must be a positive number. Also since ax is positive, we can only find the logarithm of a positive number. We will assume from now on that both are positive, but can be negative.
Find the value of x.
Note: The following identities exemplify the inverse operations of taking a power and taking a logarithm. These need to be properly understood by students.
For x > 0,
2log2 x = x.
More generally, for a > 0, x > 0,
aloga x = x.
In the other direction, for any x,
loga 2x = x.
More generally, for a > 0,
logaax = x.
It is important for students to properly understand these two general identities.
Logarithms to the base 10
You will notice that in all the examples above, the values of the logarithms were rational numbers, which were not too hard to find. Suppose we wanted to know the value of
log10 7? Thus, we seek a number x such that 7 = 10x.
We can see from the graph of y = 10x that such a number lies between 0 and 1.
The calculator is able to give an approximate value of this number. It is shown in the module, The Real Numbersthat numbers such as this are irrational.
Thus, to 4 decimal places, the calculator reports that log10 7 ≈ 0.8451.
The Logarithm Laws
Suppose a > 0 for the rest of this section.
- Law 1
- loga = 0 and loga a = 1
since a0 = 1, we have loga 1 = 0.
Similarly, since a1 = a, we have logaa = 1.
- Law 2
- If x and y are positive numbers, then logaxy = logax + logay
That is, the logaithm of a product is the sum of the logarithms.
Suppose x = ac and y = ad so that logax = c and logay = d.
|Then||xy||= ac × ad|
|=ac+d||(by Index law 1)|
|= c + d|
|= logax + logay|
- Law 3
- If x and y are positive numbers, then loga = logax − logay.
That is, the logarithm of a quatient is the difference of their logarithms.
Suppose x = ac and y = ad so that logax = c and logay = d.
- Law 4
- If x is a positive number, then loga = −logax.
This follows from logarithm law 3 and logarithm law 1.
|loga||= loga 1 − logax||(logarithm law 3)|
|= 0 − logax||(logarithm law 1)|
|= −logax, as required.|
- Law 5
- If x is a positive number and n is any rational number, then loga (xn) = nlogax.
This follows from logarithm law 3 and logarithm law 1.
|loga (xn)||= loga ((ac)n)|
|= loga (acn)||(by Index law 3)|
|= nlogax, as required.|
a logbx2 + logbx3 − logbx4b logk + logk
c logb (x2 − a2) − logb (x − a), if x > a
Change of base
Some calculators are able to find the logarithm of a number to any positive base. This is not, however, universal, and there are many occasions when we would like to change from one base to another.
For example, to find log3 7 we can change from base 3 to base 10, where the calculator can be used. Change of base is also important in calculus, where logarithms to the base are used.
The change of base formula states that:
Here is a proof of this result.
Let x = logbc, then c = bx.
Take logarithms to the base of both sides, then
logac = logabx = xlogab (using logarithm law 5).
Hence x = . That is logb c = .
Calculate log7 8 to four decimal places.
Changing form base 7 to base 10.
As a check, with the calculator, 71.0686 ≈ 7.9997.
The Logarithm graph
As we did for exponentials, we can draw the graph of y = log2x by drawing up a table of values.
We note the following features of the graph:
- the graph is to the right of the y-axis, since only logarithms of positive numbers are defined.
- as becomes small, the y values become large negative numbers. Thus, the graph approaches, but does not touch, the negative axis. We say that the negative y-axis is
an asymptote of the graph.
- the x-intercept is (1, 0) since log2 1 = 0.
- the graph does not have a y-intercept.
- as x takes large positive values, log2x becomes large.
Logarithms and exponentials are inverses of each other. Their graphs are reflections of each other in the line y = x.
This is illustrated in the following graphs of y = log3x and y = 3x.
Use a table to draw on the same diagram the graphs of y = log2x and y = log3x. What can you say about the graphs when x < 1 and x > 1 when ?
Using Logarithms to solve exponential equations
We will conclude this module with some further applications of exponentials and logarithms.
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