Log Calculator

Calculate common logarithms (log₁₀) and natural logarithms (ln) instantly with our free online Log Calculator. Perfect for students, engineers, scientists, programmers and anyone working with mathematics.

✔ log₁₀ & ln • ✔ Accurate logarithms • ✔ Instant calculation • ✔ Free calculator

What is a Log Calculator?

A Log Calculator calculates logarithms, which are the inverse of exponentiation. Instead of asking "What is 10²?", a logarithm asks "What power must 10 be raised to in order to produce 100?"

This calculator supports both common logarithms (base 10) and natural logarithms (base e), making it suitable for mathematics, engineering, finance, physics and computer science.

What is a Logarithm?

A logarithm tells you which exponent produces a given number.

For example:

• 10² = 100 → log₁₀(100) = 2
• e³ ≈ 20.0855 → ln(20.0855) ≈ 3

Types of Logarithms

Type	Base	Common Uses
Common Logarithm	10	General mathematics, engineering
Natural Logarithm	e ≈ 2.71828	Science, finance, calculus
Binary Logarithm	2	Computer science

Worked Examples

Example 1 — log₁₀(100)

• 100 = 10²
• Answer = 2

Example 2 — log₁₀(1000)

• 1000 = 10³
• Answer = 3

Example 3 — ln(e)

• e¹ = e
• Answer = 1

Example 4 — ln(20.0855)

• e³ ≈ 20.0855
• Answer ≈ 3

Where Are Logarithms Used?

Everyday Uses of Logarithms

• Richter earthquake magnitude scale.
• Sound intensity measured in decibels.
• pH scale in chemistry.
• Population and bacterial growth.
• Compound interest calculations.
• Scientific notation.

Quick Reference Table

Number	log₁₀(x)	ln(x)
1	0	0
10	1	2.302585
100	2	4.605170
1000	3	6.907755
10000	4	9.210340

Things to Remember

• Logarithms are only defined for positive numbers.
• log₁₀(1) = 0
• ln(1) = 0
• logarithms are the inverse of exponentiation.
• Choosing the correct base is important.

Common Mistakes

• Trying to calculate log(0).
• Using negative numbers.
• Confusing log with exponentiation.
• Selecting the wrong logarithm base.
• Mixing log₁₀ and ln.

Frequently Asked Questions

What is the difference between log and ln?

log usually means base 10, while ln uses the mathematical constant e (≈2.71828).

Can I calculate log(0)?

No. The logarithm of zero is undefined.

Can I calculate log of a negative number?

Not in the real number system.

Why are logarithms important?

They simplify calculations involving exponential growth, decay and very large or small numbers.

Where are logarithms used?

Mathematics, engineering, finance, chemistry, physics, biology and computer science.

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