Logarithm: Concepts with Examples | Mathematics (Maths) for JEE Main & Advanced PDF Download

Introduction

In this unit, we will study logarithms. Before exploring them, it is essential to revise the concept of indices (or exponents), as logarithms are closely linked to indices. A strong understanding of indices will help in grasping logarithmic principles effectively.

For instance, we know that:

$16=2\mathrm{<sup\; style="font-size:12px\; !important">4</sup>}16\; =\; 2^4$

Here, 4 is known as the power, exponent, or index, and 2 is called the base.

Another example is:

$64=8\mathrm{<sup\; style="font-size:12px\; !important">2</sup>}$

In this expression, 2 is the power or exponent, and 8 is the base.

Why do we study logarithms ?

To understand the need for logarithms, let’s begin with an example.

We know that:

$16=2\mathrm{<sup\; style="font-size:12px\; !important">4</sup>}$^and and  8 = 2³

Now, suppose we want to multiply 16 by 8.
We could perform the multiplication directly to get:

$16\times 8=12816\; \backslash times\; 8\; =\; 128$

However, when dealing with very large numbers, such direct calculations can become quite lengthy and difficult.

Let’s see how we can use powers (indices) instead.

$16\times 8=2\mathrm{<sup\; style="font-size:12px\; !important">4</sup>}\times 2\mathrm{<sup\; style="font-size:12px\; !important">3</sup>}=2\mathrm{<sup\; style="font-size:12px\; !important">4</sup>}$⁺³=2⁷=12816 \times 8 = 2^4 \times 2^3 = 2^{4+3} = 2^7 = 128

Using the laws of indices, we replaced the multiplication of numbers with the addition of exponents (4 + 3 = 7).

Similarly, if we divide 16 by 8, we get:

$16÷8=2\mathrm{<sup\; style="font-size:12px\; !important">4</sup>}÷2\mathrm{<sup\; style="font-size:12px\; !important">3</sup>}=2\mathrm{<sup\; style="font-size:12px\; !important">4</sup>}$⁻³=2¹=2

Here, division has been replaced by subtraction of powers (4 − 3 = 1).

If we had a simple table of powers of 2, we could easily look up $27=1282^7\; =\; 128$27=128 without doing the actual multiplication.

This idea — that multiplication can be converted into addition and division into subtraction using powers — inspired John Napier (1550–1617) and Henry Briggs (1561–1630) to develop the concept of logarithms. Logarithms thus provided a simpler way to handle large and complex calculations long before calculators and computers existed.

What is a logarithm ?

Consider the expression 16 = 2⁴. Remember that 2 is the base, and 4 is the power. An alternative, yet equivalent, way of writing this expression is log₂16 = 4. This is stated as ‘log to base 2 of 16 equals 4’. We see that the logarithm is the same as the power or index in the original expression. It is the base in the original expression which becomes the base of the logarithm.
The two statements

• 16 = 2⁴
• log₂ 16 = 4

are equivalent statements. If we write either of them, we are automatically implying the other.
Example

• If we write down that 64 = 8² then the equivalent statement using logarithms is log₈64 = 2.

Example

• If we write down that log₃27 = 3 then the equivalent statement using powers is 3³ = 27.

So the two sets of statements, one involving powers and one involving logarithms are equivalent.In the general case we have:

Key Point
if x = aⁿ then equivalently log_a x = n
log_a a = 1

Let us develop this a little more.
Because 10 = 10¹ we can write the equivalent logarithmic form: log₁₀ 10 = 1.Similarly, the logarithmic form of the statement 2¹ = 2 is log₂2 = 1.
In general, for any base a, a = a¹ and so log_a a = 1
We can see from the Examples above that indices and logarithms are very closely related. In the same way that we have rules or laws of indices, we have laws of logarithms. These are developed in the following sections.

The first law of logarithms

Suppose
x = aⁿ and y = a^m
then the equivalent logarithmic forms are
log_a x = n and log_a y = m .....(1)
Using the first rule of indicesxy = aⁿ× a^m = a^n+m
Now the logarithmic form of the statement xy = a^n+m is log_a xy = n + m. But n = log_a x and m = log_a y from (1) and so putting these results together we have
log_a xy = log_a x + log_a y
So, if we want to multiply two numbers together and find the logarithm of the result, we can do this by adding together the logarithms of the two numbers. This is the first law.

Key Point
log_a xy = log_a x + log_a y

The second law of logarithms

Suppose x = aⁿ, or equivalently log_a x = n. Suppose we raise both sides of x = aⁿ to the power m:
x^m = (aⁿ)^m
Using the rules of indices we can write this as
x^m = a^nm
Thinking of the quantity x^m as a single term, the logarithmic form is
log_a x^m = nm = mlog_a x
This is the second law. It states that when finding the logarithm of a power of a number, this can be evaluated by multiplying the logarithm of the number by that power.

Key Pointlog_a x^m = mlog_a x

The third law of logarithms

As before, suppose
x = aⁿ and y = a^m
with equivalent logarithmic forms
log_a x = n and log_a y = m ....(2)
Consider x ÷ y.
using the rules of indices.In logarithmic form
which from (2) can be writtenThis is the third law.

Key Point

The logarithm of 1

Key Point
log_a 1 = 0

The logarithm of 1 in any base is 0.

Logarithmic Formulas Summarized:

• log_b(mn) = log_b(m) + log_b(n)
• log_b(m/n) = log_b (m) – log_b (n)
• log_b (xy) = y log_b(x)
• m log_b(x) + n log_b(y) = log_b(x^myⁿ)

Examples

Example2: Suppose we wish to find log₈ 1/64
This is the same as being asked ‘what is 1/64 expressed as a power of 8 ?’
Now 1/64 can be written 64⁻¹. Noting also that 8² = 64 it follows that
using the rules of indices. So log₈ = 1/64 = -2

Example3: Suppose we wish to find log₅ 25.
This is the same as being asked ‘what is 25 expressed as a power of 5 ?’
Now 5² = 25 and so log₅ 25 = 2.

Example4: Suppose we wish to find log₂₅ 5.
This is the same as being asked ‘what is 5 expressed as a power of 25 ?’
We know that 5 is a square root of 25, that is 5 = √25. So 25 ^1/2 = 5 and so log₂₅ 5 = 1/2.

Notice from the last two examples that by interchanging the base and the number
This is true more generally:

Key Point

Example5: Consider log₂ 8. We are asking ‘what is 8 expressed as a power of 2 ?’ We know that 8 = 2³ and so log₂ 8 = 3.
What about log₈ 2 ? Now we are asking ‘what is 2 expressed as a power of 8 ?’ Now 2³ = 8 and so 2 = ∛8 or 8^1/3. So log₈ 2 =1/3.We see again

Standard bases

There are two bases which are used much more commonly than any others and deserve special mention. These are:
base 10 and base eLogarithms to base 10, log₁₀, are often written simply as log without explicitly writing a base down. So if you see an expression like log x you can assume the base is 10. Your calculator will be pre-programmed to evaluate logarithms to base 10. Look for the button marked log.
The second common base is e. The symbol e is called the exponential constant and has a value approximately equal to 2.718. This is a number like π in the sense that it has an infinite decimal expansion. Base e is used because this constant occurs frequently in the mathematical modelling of many physical, biological and economic applications. Logarithms to base e, log_e, are often written simply as ln. If you see an expression like ln x you can assume the base is e. Such logarithms are also called Naperian or natural logarithms. Your calculator will be pre-programmed to evaluate logarithms to base e. Look for the button marked ln.

Key Point
Common bases:
where e is the exponential constant. Useful results: log 10 = 1, ln e = 1

Using logarithms to solve equations

We can use logarithms to solve equations where the unknown is in the power. Suppose we wish to solve the equation 3^x = 5. We can solve this by taking logarithms of both sides. Whilst logarithms to any base can be used, it is common practice to use base 10, as these are readily available on your calculator. So, log 3^x = log 5
Now using the laws of logarithms, the left hand side can be re-written to give
x log 3 = log 5
This is more straight forward. The unknown is no longer in the power. Straightaway
If we wanted, this value can be found from a calculator.