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The equation 2³ = 8 means that 3 is the index of the power to which we raise the number 2 to produce 8.

A logarithm is an index, and in this example, 3 is the logarithm of 8 to the base 2. We write this as

Log₂ 8 = 3

These two equations are identical: 2³ = 8 and log₂ 8 = 3

They express the same fact in the language of logarithms.

DEFINITION OF A LOGARITHM

If a number y can be written in the form a^x, then the index x is called the logarithm of y to the base a.

If y = a^x

Then log_a y = x

Since 2 = 2¹, then log₂ 2 = 1

Similarly, log₃ 3 = 1, and log₁₀ 10 = 1, and as a = a¹, then log_a a = 1

Also, since 1 = 2⁰ then log₂ 1 = 0

Similarly, log₃ 1 = 0 and log₁₀ 1 = 0, and as 1 = a^7deg;, then log_a 1 = 0

Why do we use logarithms? Before the days of the electronic calculator, logarithms were used for multiplication and division and involved the use of log tables. Nowadays, logarithms are mainly used in integration or to find a linear function from an exponential one.

There are three rules which, with the definition of a logarithm, can be deduced from the rules for indices.

Let log_a x = m and log_a y = n

This means that x = a^m and y = aⁿ

So xy = a^m × aⁿ = a(^m + n)

Applying the definition of a logarithm gives

For example, log_a 21 = log_a 7 + log_a 3

Let log_a x = m and log_a y = n with x = a^m and y = aⁿ

Applying the definition of a logarithm gives

For example, log_a 7 = log_a 14 − log_a 2

Let log_a x = n with x = aⁿ

Raising each side to the power r gives

Applying the definition of a logarithm gives

For example, log₁₀ 1000 = log₁₀ 10³ = 3 log₁₀ 10 = 3 (because log₁₀ 10 = 1)

Natural logarithms

The most frequently used bases for logarithms are 10 and the number ‘e?. Rather like , the irrational number ‘e? occurs frequently in many branches of mathematics and its applications to science and engineering. Logarithms to base 10 are known as common logarithms and those to base ‘e? are called natural or Napierian logarithms after the mathematician who discovered them.

Natural logarithms have the property that log_e e^x = x.

We use natural logarithms to solve equations that contain the exponential function e^x where e is the irrational number 2.718281828 correct to ten significant figures.

When we work with logarithms to base 10 we drop the subscript and just write log x. Natural logarithms are written as 1n x. This means that the rules for logarithms can be written as follows for natural logarithms.

Change of base

The final thing that we need to be able to do with logarithms is change the base.

Suppose we have log_a x and we want to find log_b x.

Taking logarithms to base a gives

Rearranging this, gives

So,

Thus, if we want to change between natural logarithms and logarithms to the base 10, we can use

Activity 19

Simplify log 6 + log 3 − log 9
Write 4 log log y + 3 log z as a single logarithm
log 64 ÷ log 2
(log 27 − log 9) ÷ log 3
Find log₉ × given that log₃ x = 12

Answer

Discussion

Note: log₃ 3² = 2 log₃ 3 and log₃ 3 = 1.

Activity 20

Solve the equations:

15 = 3e^2x
2e^−x/10 + 16 = 20

Answer

Discussion

Activity 21

Express the following as the logarithm of one number:

Answer

Discussion

(a) Since a = a¹ then log_a a = 1, so log 10 = 1; also 1000 = 10³, so log 1000 = 3. Thus

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If you multiply out the bracket first you will still get the same answer.

Note: The value of the bracket must be evaluated from left to right; if you multiply 4 × 3 first you will not get the correct answer. This is because