A logarithm is the exponent required to raise a base to a given positive number: if b^y=x, then log_b(x)=y.

What is the change-of-base formula?

For a valid base b and any convenient base k, log_b(x)=ln(x)/ln(b)=log_k(x)/log_k(b).

Can the logarithm of zero or a negative number be calculated as a real number?

No. For real logarithms the argument must be positive, so x must be greater than zero.

Math · Exponents & logarithms

Logarithm calculator

Q: What is the difference between log and ln?

In common mathematical usage, log often means base 10 while ln means the natural logarithm with base e, where e is approximately 2.718281828.

Calculate log_b(x), common log, natural log, and antilog. See the change-of-base formula, nearby powers, and how the answer moves on the logarithmic scale.

Report error

log_b(x)

—

change of base

Common · Natural

—

log₁₀(x) · ln(x)

Antilog

—

b^log_b(x) = x

Logarithm curve — and your exact point

The graph changes with the selected base. The highlighted point is (x, log_bx).

Live

y = log_b(x)Lower powerUpper power

LIVE EXPONENT CHECK

Where does x sit between powers of the base?

Updates live

Log value—the exponent required to reach x

Lower power—b raised to the lower integer

Upper power—b raised to the next integer

Position—fractional distance between powers

Enter valid values to see how x sits between consecutive powers of b.

bᵏxbᵏ⁺¹

Nearby powers of the selected base

Live table

Exponent k	b^k	Relation to x

Log values

Argument x

The argument must be greater than 0.

Base b

The base must be positive and not equal to 1.

Quick base

Concept

A logarithm is an exponent with a question mark around it

If b^y = x, then log_b(x) = y. The calculator is solving for the missing exponent.

Base

The number being raised to a power.

Argument

The positive number inside the logarithm.

Exponent

The answer the logarithm is looking for.

Natural base

Approximately 2.718281828.

For real logarithms, the argument must be positive and the base must be positive and different from 1. Base 10 is commonly called the common logarithm; base e is the natural logarithm, written ln.

Formulas

Core logarithm formulas

These identities explain nearly every hand calculation you make with logs.

Definition

log_b(x) = y ⇔ b^y = x

A logarithm asks for the missing exponent.

Change of base

log_b(x) = ln(x) / ln(b)

Use the same convenient base in numerator and denominator.

Product

log_b(xy) = log_b(x) + log_b(y)

Multiplication becomes addition.

Quotient

log_b(x/y) = log_b(x) − log_b(y)

Division becomes subtraction.

Power

log_b(xⁿ) = n log_b(x)

An exponent can move to the front as a multiplier.

Reciprocal

log_b(1/x) = −log_b(x)

Taking a reciprocal changes the sign.

b^log_b(x) = x and log_b(b^y) = y

Examples

Worked logarithm examples

Use these as patterns when you want to check a result without relying on a calculator.

Example 01 · common log

log₁₀(1000)

Ask: “10 to what power gives 1000?” Since 10³ = 1000, the answer is immediate.

log₁₀(1000) = 3

Example 02 · base 2

log₂(100)

Using change of base: log(100) / log(2) = 2 / 0.30103.

log₂(100) ≈ 6.643856

Example 03 · natural log

ln(e⁵)

The logarithm and exponential undo one another when they use the same base.

ln(e⁵) = 5

Example 04 · negative result

log₁₀(0.01)

Because 10⁻² = 0.01, the exponent is negative.

log₁₀(0.01) = −2

Common logarithm values

Reference

x	log₁₀(x)	ln(x)	Useful observation
0.01	−2	−4.605170	Below 1 → negative logarithm.
0.1	−1	−2.302585	10⁻¹ = 0.1.
1	0	0	Every valid base raised to 0 is 1.
2	0.301030	0.693147	Useful for change-of-base work.
10	1	2.302585	10¹ = 10.
100	2	4.605170	10² = 100.
1000	3	6.907755	10³ = 1000.

Reference tables

Choose the base that matches the problem

Different bases are useful in different contexts. The mathematics is the same; the scale changes.

Common logarithm bases

Quick reference

Notation	Base	Typical context	Identity
log₁₀(x)	10	Scientific notation, decimal orders of magnitude	10ʸ = x
ln(x)	e ≈ 2.718281828	Calculus, continuous growth, natural models	eʸ = x
log₂(x)	2	Computer science, binary scaling	2ʸ = x

Special cases to remember

Exam check

Expression	Result	Why
log_b(1)	0	b⁰ = 1
log_b(b)	1	b¹ = b
log_b(bⁿ)	n	The logarithm retrieves the exponent.
log_b(0)	undefined	There is no real exponent that makes bʸ equal zero.
log_b(negative)	not real	The real logarithm requires a positive argument.

Applications

Where logarithms appear outside the classroom

Logarithms are useful when values span many orders of magnitude or when multiplication and exponential growth need to be expressed on a manageable scale.

Acidity

A logarithmic scale used for hydrogen-ion concentration.

Sound

Decibels express ratios on a logarithmic scale.

2ⁿ

Computing

Binary logarithms help describe powers of two and algorithmic growth.

Growth

Natural logarithms arise naturally in continuous growth and calculus.

Examples of logarithmic measurement include pH, decibels and several scientific scales. In mathematics and computing, logarithms also turn multiplicative relationships into additive ones, which makes many problems easier to analyze.

Problem solving

How to solve logarithm equations step by step

The most useful skill is not pressing a button; it is knowing which form to turn the equation into.

Type A · direct definition

log₂(x) = 5

Rewrite the equation exponentially: 2⁵ = x.

x = 32

Type B · change of base

log₃(50)

Use ln(50) / ln(3). This is the standard way to evaluate an arbitrary base with a basic calculator.

log₃(50) ≈ 3.560877

Type C · combine logs

log₁₀(2) + log₁₀(5)

Use the product rule: log₁₀(2 × 5) = log₁₀(10).

= 1

Type D · power rule

log₅(25²)

Bring the exponent forward: 2 log₅(25) = 2 × 2.

= 4

Common mistake

log(x + y) ≠ log(x) + log(y)

The product rule applies to multiplication, not addition.

Domain check

log_b(x) requires x > 0

Any algebraic solution that makes a log argument zero or negative must be rejected in the real-number setting.

Why logarithms matter

From hand calculation to modern science

Logarithms were developed as a way to make difficult multiplication, division and powers more manageable. Today they remain useful because many physical and mathematical relationships span several orders of magnitude.

1614

Napier

John Napier published influential logarithm tables.

log

Scale

Multiplicative changes become additive distances.

Calculus

The natural logarithm has derivative 1/x.

dB · pH

Applications

Logarithmic scales remain common in science and engineering.

FAQ

Logarithm calculator questions

What is a logarithm?

A logarithm is the exponent needed to raise a base to an argument. If b^y = x, then log_b(x) = y.

What is the difference between log and ln?

In common mathematical usage, log often denotes base 10, while ln denotes the natural logarithm with base e ≈ 2.718281828. Always check the convention used by the problem.

How do I calculate a logarithm with a base other than 10 or e?

Use change of base: log_b(x) = ln(x) / ln(b). The same result can be obtained with common logs: log_b(x) = log₁₀(x) / log₁₀(b).

Can a logarithm be negative?

Yes. A positive argument between 0 and 1 has a negative logarithm for bases greater than 1. For example, log₁₀(0.01) = −2.

Why can't I enter zero as x?

For real logarithms, x must be greater than zero. There is no real exponent y for which a valid positive base b produces b^y = 0.