Logarithms: Definition, Properties, Rules and Examples

Logarithms are an important part of mathematics because they connect exponents with numbers. A logarithm tells us the power to which a base must be raised to get a given number. In simple words, logarithm answers this question: “What power is needed?”

For example, 10³ = 1000. So log₁₀1000 = 3. This means the logarithm of 1000 to base 10 is 3 because 10 must be raised to power 3 to get 1000.

Logarithms are used in algebra, scientific calculations, compound growth, sound intensity, pH value, computer science, data scale, earthquake measurement, and competitive exam questions. They make multiplication, division, and powers easier by changing them into addition, subtraction, and multiplication.

This article explains logarithms in simple language. The existing formulas and images in this post are kept as they are, and new explanations, rules, examples, tables, solved questions, and FAQs are added to make the topic complete.

What Is a Logarithm?

A logarithm is the exponent or power to which a base is raised to obtain a number.

If a^m = x, then log_ax = m.

Here a is the base, x is the number, and m is the logarithm.

Example: 2⁵ = 32, so log₂32 = 5.

Example: 3⁴ = 81, so log₃81 = 4.

Example: 5² = 25, so log₅25 = 2.

Logarithmic Form and Exponential Form

Logarithmic form and exponential form are two ways to write the same relation.

Exponential form	Logarithmic form
2³ = 8	log₂8 = 3
10⁴ = 10000	log₁₀10000 = 4
7² = 49	log₇49 = 2
4⁰ = 1	log₄1 = 0

To convert exponential form into logarithmic form, remember this pattern: base stays base, answer becomes log input, and power becomes log value.

Existing Explanation and Formulas

A logarithm is a mathematical tool denoting the exponent a base requires to yield a specific number. Noted as “log_b(x),” where “b” is the base and “x” the value, logarithms simplify calculations by transforming multiplication and exponentiation into addition, playing a pivotal role in various fields for their efficiency in handling complex relationships.

It is a passive real number other than $a^m = n$
We write $m=log_a x$

$10^3 =1000$ 4 then we can write $log_{10} 1000 = 3$
$2^{-3} =\frac{1}{8}$ then we can write $log_2 \frac{1}{8} = -3$

Properties of logarithms

Multiplicative Property: log_b(xy) = log_b(x) + log_b(y)
Divisional Property: log_b(x/y) = log_b(x) – log_b(y)
Power Rule: log_b(x^n) = n * log_b(x)
Change of Base Formula: log_b(x) = log_a(x) / log_a(b)
Logarithm of 1: log_b(1) = 0
Logarithm of Base: log_b(b) = 1
Negative Number Logarithm: log_b(x) is undefined for x ≤ 0
Logarithm of Infinity: log_b(∞) = ∞
Logarithm of Fraction: log_b(1/x) = -log_b(x)
Logarithmic Identity: b^(log_b(x)) = x

$log_a (xy) = log_a x +log_a y$
$log_a (\frac{x}{y}) = log_a x - log_a y$
$log_x x = 1$
Example $2^1=2$ so we can write $log_2 2 = 1$
$log_a 1 = 0$
Example $5^0=1$ in logarithm form $log_5 1 = 0$

$log_a x^p = p(log_a x)$
$log_a x = 1/log_x^a$
$log_a x = \frac{log_b^x}{log_b^a} = \frac{log_⁡x}{log_⁡a}$

Logarithms to the base 10 are known as common logarithms. When base is not mentioned it is taken as 10.
Characteristic – when the number is greater than 1 – the characteristic is one less from the number of digit in the left of the decimal point in the given number.

$(\overline{48}) ̅.48= 1$ $(\overline{6185})$ ̅.41 = 3
When the number is less than 1 one more than the number of zero between the decimal point & the 1st significant digit of the number & it is negative.
0.518 = -1 0.0347 = etc.

Log table-
$10^{.001} = 10^{1/1000} = 1.002305.238$
$10^{0.02} = 1.584293192$
$10^{0.01} = 10^{1/100} = 1.023292992$
$10^{0.6} =3.981071706$
$10^{0.5} = 10^{\frac{1}{2}}= 3.16227766$
$10^{0.3} = 1.995262215$
$10^{0.1} = 10^{\frac{1}{10} = 1.258925412$
$10^{0.7} = 5.0118872336$

Conditions for Logarithms

For log_ax to be defined in real numbers, some conditions must be followed.

The base a must be positive.
The base a must not be equal to 1.
The number x must be positive.

So log₂8 is defined, but log₁8 is not defined. Also, log of 0 and log of a negative number are not defined in real-number logarithms.

Common Logarithm

A logarithm with base 10 is called a common logarithm. It is written as log₁₀x. In many school books and calculators, if the base is not written, log x usually means log₁₀x.

Examples:

log 10 = 1 because 10¹ = 10.
log 100 = 2 because 10² = 100.
log 1000 = 3 because 10³ = 1000.
log 1 = 0 because 10⁰ = 1.

Natural Logarithm

A logarithm with base e is called a natural logarithm. It is written as ln x. The number e is an important mathematical constant whose approximate value is 2.71828.

Natural logarithms are widely used in higher mathematics, calculus, growth and decay problems, compound interest, and science. For basic aptitude, common logarithms and log rules are usually more important, but knowing ln helps when you move to advanced topics.

Basic Properties of Logarithms

Property	Formula
Log of product	log_a(xy) = log_ax + log_ay
Log of quotient	log_a(x/y) = log_ax – log_ay
Power rule	log_a(xⁿ) = n log_ax
Log of 1	log_a1 = 0
Log of base	log_aa = 1
Inverse rule	a^log_ax = x
Change of base	log_ax = log_bx / log_ba

Product Rule of Logarithms

The logarithm of a product is equal to the sum of the logarithms of the factors.

log_a(xy) = log_ax + log_ay

Example: log₁₀(2 × 5) = log₁₀2 + log₁₀5.

Since 2 × 5 = 10, log₁₀10 = 1. So log 2 + log 5 = 1.

Quotient Rule of Logarithms

The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator.

log_a(x/y) = log_ax – log_ay

Example: log(100/10) = log100 – log10.

log100 = 2 and log10 = 1.

So log(100/10) = 2 – 1 = 1.

Power Rule of Logarithms

The logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number.

log_a(xⁿ) = n log_ax

Example: log₂(8²) = 2 log₂8.

Since log₂8 = 3, the answer is 2 × 3 = 6.

Change of Base Formula

The change of base formula helps us calculate logarithms in a different base.

log_ax = log_bx / log_ba

Example: log₂8 = log 8 / log 2.

Using common log values, log 8 = 0.9030 and log 2 = 0.3010.

log₂8 = 0.9030 / 0.3010 = 3.

Logarithm of 1 and Logarithm of Base

For any valid base a, log_a1 = 0 because a⁰ = 1.

Example: log₅1 = 0.

For any valid base a, log_aa = 1 because a¹ = a.

Example: log₇7 = 1.

Logarithm of a Reciprocal

The logarithm of a reciprocal changes the sign.

log_a(1/x) = -log_ax.

Example: log₂(1/8) = -3 because 2^-3 = 1/8.

Characteristic and Mantissa

In common logarithms, the value of a logarithm has two parts: characteristic and mantissa. The characteristic is the integer part. The mantissa is the decimal part.

Example: If log 48 = 1.6812, then 1 is the characteristic and 0.6812 is the mantissa.

For numbers greater than 1, the characteristic is one less than the number of digits before the decimal point.

Number	Digits before decimal	Characteristic
48	2	1
6185	4	3
756	3	2
92.5	2	1

For numbers less than 1, the characteristic is negative. It depends on the number of zeros after the decimal point before the first non-zero digit.

Example: For 0.518, there is no zero between decimal point and first significant digit, so characteristic is -1.

Example: For 0.0347, there is one zero between decimal point and first significant digit, so characteristic is -2.

Expanding Logarithmic Expressions

Expanding means writing one logarithm as a sum or difference of simpler logarithms.

Example: Expand log_a(xy²).

log_a(xy²) = log_ax + log_ay².

= log_ax + 2log_ay.

Example: Expand log_a(m/n).

log_a(m/n) = log_am – log_an.

Combining Logarithmic Expressions

Combining means writing a sum or difference of logarithms as one logarithm.

Example: log x + log y = log(xy).

Example: log a – log b = log(a/b).

Example: 2log x = log(x²).

Example: log 2 + log 3 – log 6 = log(2 × 3 / 6) = log1 = 0.

Solving Simple Logarithmic Equations

To solve logarithmic equations, convert the logarithmic form into exponential form or use properties of logs. Always check the final answer because the number inside a logarithm must be positive.

Example 1

Solve log₂x = 5.

Convert to exponential form.

x = 2⁵ = 32.

Example 2

Solve log₃81 = x.

3^x = 81.

81 = 3⁴.

So x = 4.

Example 3

Solve log x = 2.

Base is 10 because no base is written.

x = 10² = 100.

Example 4

Solve log₅(x + 1) = 2.

x + 1 = 5² = 25.

x = 24.

Important Values of Common Logarithms

Value	Common logarithm
log 1	0
log 2	0.3010
log 3	0.4771
log 5	0.6990
log 10	1

These values are useful in many aptitude questions. For example, if log 2 is given as 0.3010, then log 8 can be found as log(2³) = 3log2 = 0.9030.

Aptitude Examples on Logarithms

Example 1

If log 2 = 0.3010, find log 16.

16 = 2⁴.

log16 = log(2⁴) = 4log2 = 4 × 0.3010 = 1.2040.

Example 2

If log 2 = 0.3010 and log 3 = 0.4771, find log 6.

6 = 2 × 3.

log6 = log2 + log3 = 0.3010 + 0.4771 = 0.7781.

Example 3

If log 2 = 0.3010, find log(1/8).

1/8 = 2^-3.

log(1/8) = log(2^-3) = -3log2 = -0.9030.

Example 4

Find the number of digits in 2¹⁰⁰, given log2 = 0.3010.

Number of digits in N = integer part of logN + 1.

log(2¹⁰⁰) = 100log2 = 30.10.

Integer part is 30.

Number of digits = 30 + 1 = 31.

Example 5

Solve log(x + 5) + log(x – 5) = log144.

Use product rule.

log[(x + 5)(x – 5)] = log144.

x² – 25 = 144.

x² = 169.

x = 13 or -13. But x – 5 must be positive, so x must be greater than 5.

Therefore, x = 13.

Common Mistakes in Logarithms

Writing log(a + b) = loga + logb. This is wrong.
Forgetting that logarithms are defined only for positive numbers.
Using base 10 automatically when a different base is written.
Forgetting that log_a1 = 0.
Forgetting that log_aa = 1.
Not checking answers in logarithmic equations.
Confusing log x² with (log x)². These are different expressions.

Practice Questions

Convert 2⁶ = 64 into logarithmic form.
Convert log₅125 = 3 into exponential form.
Find log₁₀10000.
Find log₃27.
Simplify log 2 + log 5.
Simplify log 100 – log 10.
If log2 = 0.3010, find log32.
If log2 = 0.3010 and log3 = 0.4771, find log12.
Solve log₂x = 4.
Solve log(x – 1) = 2.
Find the number of digits in 5²⁰ if log5 = 0.6990.
Simplify log_a(x³y).

Answers to Practice Questions

log₂64 = 6.
5³ = 125.
4.
3.
log10 = 1.
log10 = 1.
log32 = log(2⁵) = 5log2 = 1.5050.
log12 = log(3 × 4) = log3 + 2log2 = 0.4771 + 0.6020 = 1.0791.
x = 16.
x – 1 = 100, so x = 101.
log(5²⁰) = 20 × 0.6990 = 13.98. Number of digits = 14.
3log_ax + log_ay.

Real Life Uses of Logarithms

Logarithms are used when numbers grow or shrink very quickly. In science, pH is measured using logarithms. Sound intensity is measured using a logarithmic scale. Earthquake magnitude also uses a logarithmic scale.

In finance, logarithms help in compound growth calculations. In computers, logarithms are used in algorithms, data structures, search problems, and information theory. In biology, population growth and bacterial growth can also be studied using exponential and logarithmic models.

The main idea is simple: logarithms help us handle very large or very small numbers in a manageable way.

Frequently Asked Questions

What is a logarithm?

A logarithm is the power to which a base must be raised to get a given number.

What is log₁₀1000?

log₁₀1000 = 3 because 10³ = 1000.

What is common logarithm?

A logarithm with base 10 is called a common logarithm.

What is natural logarithm?

A logarithm with base e is called a natural logarithm. It is written as ln x.

What is log_a1?

log_a1 = 0 for any valid base a, because a⁰ = 1.

What is log_aa?

log_aa = 1 for any valid base a, because a¹ = a.

Can we take log of a negative number?

In real-number mathematics, log of a negative number is not defined.

Can we take log of zero?

No. Logarithm of zero is not defined.

What is the product rule of logarithms?

The product rule says log_a(xy) = log_ax + log_ay.

What is the change of base formula?

The change of base formula is log_ax = log_bx / log_ba.

More Solved Examples on Logarithms

Example 1: Find log₄64

We need the power to which 4 must be raised to get 64.

4¹ = 4, 4² = 16, and 4³ = 64.

So, log₄64 = 3.

Example 2: Find log₉81

We can write 81 as 9².

So, log₉81 = 2.

Example 3: Find log₂(1/32)

We know 32 = 2⁵.

So 1/32 = 2^-5.

Therefore, log₂(1/32) = -5.

Example 4: Simplify log_am + log_an – log_ap

Use product rule for addition and quotient rule for subtraction.

log_am + log_an = log_a(mn).

Now subtract log_ap.

Answer = log_a(mn/p).

Example 5: Solve log₃(2x + 1) = 4

Convert into exponential form.

2x + 1 = 3⁴.

2x + 1 = 81.

2x = 80.

x = 40.

Check: 2x + 1 = 81, which is positive, so the answer is valid.

Example 6: Solve log(x + 2) + log(x – 1) = log20

Use product rule.

log[(x + 2)(x – 1)] = log20.

(x + 2)(x – 1) = 20.

x² + x – 2 = 20.

x² + x – 22 = 0.

This gives x = (-1 ± √89)/2.

But x – 1 must be positive, so only the value greater than 1 is accepted.

Number of Digits Using Logarithms

Logarithms are very useful for finding the number of digits in a large power without calculating the full number. If N is a positive integer, then:

Number of digits in N = integer part of logN + 1.

Example: Find the number of digits in 3²⁰, given log3 = 0.4771.

log(3²⁰) = 20log3 = 20 × 0.4771 = 9.542.

Integer part is 9.

Number of digits = 9 + 1 = 10.

Example: Find the number of digits in 7¹⁵, given log7 = 0.8451.

log(7¹⁵) = 15 × 0.8451 = 12.6765.

Integer part is 12.

Number of digits = 12 + 1 = 13.

Logarithms and Scientific Notation

Scientific notation and logarithms are closely related because both use powers of 10. A number such as 500000 can be written as 5 × 10⁵. Its logarithm is close to 5 because the number lies between 10⁵ and 10⁶.

Similarly, a decimal number such as 0.004 lies between 10^-3 and 10^-2. Its common logarithm has a negative characteristic. This helps us estimate the size of very small numbers.

How to Study Logarithms Step by Step

First learn exponents clearly because logarithms are based on powers.
Practise converting exponential form into logarithmic form.
Learn product, quotient, and power rules before attempting equations.
Remember that log works only for positive numbers in real mathematics.
Practise common log values such as log2, log3, log5, and log10.
Use the change of base formula when the base is not convenient.
Check answers after solving logarithmic equations.

Quick Revision Table

Question type	Method
Convert a^m = x	Write log_ax = m
Convert log_ax = m	Write a^m = x
log of product	Use addition
log of quotient	Use subtraction
log of power	Bring power in front
base not suitable	Use change of base formula
log equation	Convert to exponential form or combine logs
number of digits	Use integer part of common log plus 1

One-Minute Recap

A logarithm is simply a power. If 2⁶ = 64, then log₂64 = 6. Product becomes addition in logarithms, quotient becomes subtraction, and powers come in front as multiplication.

Common logarithm uses base 10. Natural logarithm uses base e. For real numbers, the number inside a logarithm must always be positive, and the base must be positive but not equal to 1.

Conclusion

Logarithms are the reverse way of thinking about exponents. If exponents tell us the result of repeated multiplication, logarithms tell us the power used to get that result.

To learn logarithms well, remember the basic conversion between exponential and logarithmic form, practise the product, quotient, and power rules, and always check the base and domain of the logarithm.

What Is a Logarithm?

Logarithmic Form and Exponential Form

Existing Explanation and Formulas

Properties of logarithms

Conditions for Logarithms

Common Logarithm

Natural Logarithm

Basic Properties of Logarithms

Product Rule of Logarithms

Quotient Rule of Logarithms

Power Rule of Logarithms

Change of Base Formula

Logarithm of 1 and Logarithm of Base

Logarithm of a Reciprocal

Characteristic and Mantissa

Expanding Logarithmic Expressions

Combining Logarithmic Expressions

Solving Simple Logarithmic Equations

Example 1

Example 2

Example 3

Example 4

Important Values of Common Logarithms

Aptitude Examples on Logarithms

Example 1

Example 2

Example 3

Example 4

Example 5

Common Mistakes in Logarithms

Practice Questions

Answers to Practice Questions

Real Life Uses of Logarithms

Frequently Asked Questions

What is a logarithm?

What is log101000?

What is common logarithm?

What is natural logarithm?

What is loga1?

What is logaa?

Can we take log of a negative number?

Can we take log of zero?

What is the product rule of logarithms?

What is the change of base formula?

More Solved Examples on Logarithms

Example 1: Find log464

Example 2: Find log981

Example 3: Find log2(1/32)

Example 4: Simplify logam + logan – logap

Example 5: Solve log3(2x + 1) = 4

Example 6: Solve log(x + 2) + log(x – 1) = log20

Number of Digits Using Logarithms

Logarithms and Scientific Notation

How to Study Logarithms Step by Step

Quick Revision Table

One-Minute Recap

Conclusion

What is log₁₀1000?

What is log_a1?

What is log_aa?

Example 1: Find log₄64

Example 2: Find log₉81

Example 3: Find log₂(1/32)

Example 4: Simplify log_am + log_an – log_ap

Example 5: Solve log₃(2x + 1) = 4