# Indexes and Bases

Consider a number written as 32.

The number 3 is called the base; the number 2 is the index (plural indices). 32 is read as three squared and means multiply two 3's together. In other words

3 × 3 = 9.

Other examples:

23 (read as 2 cubed - multiply three 2's together) = 2 × 2 × 2 = 8
34 (3 to the power of 4 - multiply four 3's together) = 3 × 3 × 3 × 3 = 81

There are four simple laws of indices that apply to numbers with the same base:

 am × an = a(m + n) (When two numbers are multiplied, their indices are added) am / an = a(m - n) (When two numbers are divided, their indices are subtracted) n√am = am / n (For the nth root divide the index by n) (am)n = am × n (Raising a number to the nth power, multiply the index by n)

Here are some examples of these laws:

• 23 × 24 = (2 × 2 × 2) × (2 × 2 × 2 × 2) = 2(3 + 4) = 27 = 128 (Adding indices)

• 35 / 32 = (3 × 3 × 3 × 3 × 3) / (3 × 3) = 3(5 - 2) = 33 = 27 (Subtracting indices)

• √44 = 4(4 / 2) = 42 = 16 (Dividing indices)

• (23)2 = 2(3 × 2) = 26 = 64 (Multiplying indices)

Note that indices make calculations simpler. These indices can be used to define Logarithms.

We can say that

3 is the logarithm of 8 to base 2 (because 23 = 8)

and

4 is the logarithm of 81 to base 3 (because 34 = 81).

These are written:

log28 = 3 and log381 = 4

# Logarithms

Between 1614 and 1624, two European mathematicians, John Napier (1550 - 1617) and Henry Briggs (1561 - 1630) worked out that logarithms could be used to simplify calculations in physics and astronomy. There is good evidence that tables of the logarithms of sines and cosines were in use by the Arab astronomer, Ibn Jounis as early as the 13th century.

Logarithms turned out to be one of the most important aids to computation before the arrival of computers and calculators. Tables of logarithms were produced to aid computation. These tables continued to be used in schools until the mid 20th Century.

Indices can be applied to any base. The tables of logarithms most useful in computations use a base of 10. These are called Common Logarithms. Any base could be used in theory. Base 10 simplifies the work involved in calculations because our number system is base 10. We can apply the laws of indices as before to base 10.

103 × 104 = 1,000 × 10,000 = 10(3 + 4) = 107 = 10,000,000

The logarithm of 1000 to base 10 is 3 (remember 103 = 1000). This is written:

log101000 = 3

Because base 10 is so important, it is assumed if no base is indicated. The above can also be written simply as

log (1000) = 3.

Note that the indices 3 and 4 (above) tell us how many zeros the numbers 1,000 and 10,000 contain.

Here is a list of some whole number base 10 logarithms.

Number Equivalent Logarithm
10,000,000
107
7
1,000,000
106
6
100,000
105
5
10,000
104
4
1,000
103
3
100
102
2
10
101
1
1
100
0

Note that the logarithm of 1 is 0. This is because 100 = 1. This makes sense. When you multiply a number by 1 you do not change its value. Correspondingly, if you add 0 to the index you leave it unchanged.

10 × 1 = 101 × 100 = 10(1 + 0) = 101 = 10

There is more on indices in my Binomial Theorem essay.

There is more (much more) to logarithms than the whole number values discussed so far. A number like 63 will have as its logarithm a number between 1 and 2. In fact, 63 can be written as

101.799340549....

so

log(63) = 1.799340549....

This number is actually transcendental and its decimal part goes on for ever. Tables of logarithms will usually list these numbers to four decimal places (or seven in better quality tables). Calculators may work out logarithms to nine or more decimal places. Using logarithms simplifies calculations but the answers will never be 100% exact except in the rare cases of whole number logarithms. But then you should not need logarithms to multiply or divide by 100, should you?

Let us see how logarithms can be used in calculations. Imagine that we wish to multipy two numbers, say, 63 and 41.

By using tables of logarithms the two numbers can be written as

101.7993 × 101.6128 (to four decimals)

The multiplication can then be done by adding the indices:

10(1.7993 + 1.6128) = 103.4121

By using a reverse logarithm table (called an anti-logarithm) this can be reduced to 2582.85. The actual answer is 2583 so we have done a multiplication to four places of accuracy by looking up some figures in a table and adding them. In the days when calculations of planetary orbits could take years this would have been a major advance.

Remember that logarithms are only as accurate as the number of decimal places used. We could have used logarithms of more decimal places to increase the accuracy.

Let us now multiply 630 by 4.1. This can be shown to be equivalent to

102.7993 × 100.6127

Notice that the decimal part of the index is the same as for the 63 × 41 case. The table below shows this:

Number Equivalent Logarithm
63,000,000
107.7993
7.7993
6,300,000
106.7993
6.7993
630,000
105.7993
5.7993
63,000
104.7993
4.7993
6,300
103.7993
3.7993
630
102.7993
2.7993
63
101.7993
1.7993
6.3
100.7993
0.7993

This only works for base 10 logarithms.

Tables of base 10 logarithms only need to give the decimal part of the logarithm for the number 63. The user can add the integer part by knowing where the decimal place is in the original number. Since 63 lies between 10 (101) and 100 (102), the logarithm must be between 1 and 2, therefore the integer part must be 1.

We can also use tables for the reverse of this process. In this case we need to find the anti-logarithm of a logarithm. For example

Find the number with a logarithm of 2.7993

Only the decimal part of the logarithm is looked up in the tables (.7993). This will give the figures 63. The integer of the logarithm (2) tells us that the value will be between 100 and 1,000.

The anti-logarithm of 2.7993 is, therefore, 630.

Let us do a division.

Evaluate 630 / 41

Again, we will use logarithms to four decimal places.

630 / 41 = 102.7993 / 101.6128 = 10(2.7993 - 1.6128) = 101.1865 = 15.36

Logarithms also give us an alternative way (to the Binomial Theorem) of calculating roots. Let us find √6300. Again, by using logarithms:

√6300 = √(103.7993) = 10(3.7993 / 2) = 10 1.89965 = 79.37.

Before moving on let us summarise the laws of logarithms. Remember, logarithms are really indices so the laws are similar to the laws of indices. These laws are the same regardless of the base.

When two numbers are multiplied, their logarithms are added:

Log (m × n) = Log m + Log n

When two numbers are divided, their logarithms are subtracted:

Log (m / n) = Log m - Log n

For the nth root of a number divide the logarithm by n:

Log (n√m) = (1 / n ) Log m

To Raise a number to the nth power, multiply the logarithm by n:

Log mn = n Log m

# Series For Logarithms

There is a series for calculating logarithms. This gives values for logarithms to base e. e has the value of 2.71828183.... Logarithms to base e are called Natural Logarithms (or Naperian Logarithms after their European discoverer). They are used in various branches of mathematics (eg. calculus) but not usually in computation.

Logarithms to base e can be written as

logeN

and more commonly as

ln N.

The series is shown below.

This infinite series only gives finite values if x is greater that -1 and less than or equal to 1:

-1 < x <= 1

For example, the natural logarithm of 1.3 can be evaluated as follows:

ln(1.3) = ln(1 + 0.3)

= (0.3) - (0.3)2/2 + (0.3)3/3 - ...

= 0.3 - 0.045 + 0.003 - ... = 0.258

Once a natural logarithm has been evaluated, it can be converted to a common logarithm.

Logarithms can be converted to different bases using the formula:

logaN = (logbN) / (logba)

To convert a natural logarithm (logeN) to a common logarithm (log10M) this formula becomes:

log10N = logeN / loge10 = (ln N) / (ln 10) = (ln N) / 2.30258 = 0.4343 × ln N

In other words, multiply a natural logarithm by 0.4343 to convert it to a common logarithm. Incidently, the two numbers that occur in the above line have the following derivation:

0.4343 = log10e and 2.30258 = loge10 (ln 10)

So, if ln(1.3) = 0.258 (from the logarithm series above), the value of log101.3 can be obtained by:

log101.3 = 0.4343 × ln(1.3) = 0.4343 × 0.258 = 0.1120

This implies values like log 13 = 1.1120, log 130 = 2.1120, etc. This is how tables of logarithms were originally calculated.

# Solving Equations With Logarithms

Logarithms can be used to solve algebraic equations where the unknown is in the index. For example:

Example 1: Solve the equation 2x = 10.

This is best done by taking logarithms on both sides, the base does not matter as long as we are consistant. In these examples, we use logarithms to base 10.

2x = 10 becomes Log 2x = Log 10

The left hand side can be rearranged using the laws of logarithms:

Log 2x = Log 10 becomes x × Log 2 = Log 10 which evaluates to x × 0.3010 = 1.

The value of x is 3.322.

We can check this answer by putting the value of x back into the original equation to get

23.322 = 10.000.

Example 2: Solve 3x - 1 = 52x+3.

Take Logarithms on both sides: Log 3x - 1 = Log 52x+3

This rearranges to (x - 1) × Log 3 = (2x + 3) × Log 5

Evaluating the logarithms gives: (x - 1) × 0.4771 = (2x + 3) × 0.6990

The rest is just simple algebra (if a bit messy):

0.4771x - 0.4771 = 1.398x + 2.097,

therefore x = -2.795 (to three decimals).

This can also be checked by setting x to this value in the original equation.

This gives 3-3.795 = 5-2.59 which yields 0.01546 = 0.01547,

which is correct to four decimal places.

Equations may already have a logarithm. For example,

Example 3: Solve the equation: Log (x2 + 1) = 1

Since there is already a logarithm on the left hand side we "anti-log" both sides. That gets rid of the logarithm on the left and makes the right hand side an index of 10 (see the original definition of a logarithm).

Log (x2 + 1) = 1 becomes x2 + 1 = 101 which is simply x2 + 1 = 10

This is now a standard quadratic equation: x2 - 9 = 0 with two answers, x = 3 and x = -3.

# KryssTal Related Pages

More on series. Introduction to different types of numbers: Real, Imaginary, Rational, Irrational, Transcendental.

A series devised by Isaac Newton that is used for calculations. More on indices: roots and powers. Factorials. Combinations.

Right-angled triangles, Sines, Cosines, Tangents. Using trigonometric Functions, series and formulas.