A logarithm answers: "To what exponent must I raise the base to get this number?" log₁₀(100) = 2 because 10² = 100. It is the inverse of exponentiation.

What is the natural logarithm?

The natural logarithm (ln) uses base e ≈ 2.71828. It appears naturally in calculus, probability, and continuous growth/decay formulas.

What is the change of base formula?

log_b(x) = ln(x)/ln(b) or log_b(x) = log(x)/log(b). This lets you compute any base logarithm using just natural or common logs.

Why can't I take the log of zero or negative numbers?

In real numbers, the logarithm is only defined for positive values. There is no real exponent that makes a positive base equal zero or a negative number.

How are logarithms used in computer science?

Binary logarithms (log₂) measure algorithm complexity. Binary search takes O(log₂ n) steps. Information entropy is measured in bits using log₂.

What is the relationship between log and exponential?

They are inverse functions. If y = b^x, then x = log_b(y). This means log_b(b^x) = x and b^(log_b(x)) = x.

Logarithm Calculator

Calculate logarithms of any base. Compute log_b(x) = ln(x)/ln(b) for common, natural, and custom base logarithms.

Number (x)Must be > 0

Base

Compare: log(x2)Second value to compare

log_10(1000)

Exact integer: 10^3 = 1000

log10(x)

Common logarithm -- order of magnitude: 10^2

ln(x)

6.907755279

Natural log -- e^6.907755 = 1000

log2(x)

9.9657842847

Binary log -- 1000 needs 10 bits

Floor / Ceil

2 / 3

10^2 = 100 <= 1000 <= 1000 = 10^3

d/dx log_b(x)

4.342945e-4

1 / (x * ln(b)) = 1 / (1000 * 2.3026)

Verification

1000

b^(log_b(x)) should equal x = 1000

log(500)

2.6989700043

Difference: 0.30103 -- ratio of x values: 2

Logarithmic Scale Position (log10)

10^-310^9

x = 1000, log10 = 3

Antilog (Inverse) Calculator

Compute b^y -- the inverse of logarithm

Exponent (y) -- compute 10^y

10^3 = 1,000.000000

Logarithm Identities Applied to x = 1000

Identity	Expression	Value
Product rule	log(x*10) = log(x) + log(10)	4
Quotient rule	log(x/10) = log(x) - log(10)	2
Power rule	log(x^2) = 2*log(x)	6
Reciprocal	log(1/x) = -log(x)	-3
Square root	log(sqrt(x)) = log(x)/2	1.5
Change of base	log2(x) = ln(x)/ln(2)	9.96578428

Multi-Base Logarithm Reference

x	log10	ln	log2	log_10
0.001	-3	-6.907755	-9.965784	-3
0.01	-2	-4.60517	-6.643856	-2
0.1	-1	-2.302585	-3.321928	-1
0.5	-0.30103	-0.693147	-1	-0.30103
1	0	0	0	0
2	0.30103	0.693147	1	0.30103
e	0.434294	1	1.442695	0.434294
pi	0.49715	1.14473	1.651496	0.49715
5	0.69897	1.609438	2.321928	0.69897
10	1	2.302585	3.321928	1
100	2	4.60517	6.643856	2
1000	3	6.907755	9.965784	3
10^4	4	9.21034	13.287712	4
10^6	6	13.815511	19.931569	6
10^9	9	20.723266	29.897353	9

Real-World Logarithmic Scales

Scale	Formula	Interpretation for x = 1000
Decibels (sound)	10 * log10(ratio)	30 dB
pH (acidity)	-log10([H+])	pH = -3
Richter (earthquake)	log10(amplitude)	Magnitude 3
Bits (info theory)	log2(states)	9.97 bits
Doubling time	ln(2) / rate	0.1003 (if x is rate)

Planning notes, formulas, and examples

About the Logarithm Calculator

The Logarithm Calculator computes the logarithm of any positive number to any positive base. Logarithms answer the question: "To what power must I raise the base to get x?" If b^y = x, then log_b(x) = y.

This calculator supports all common bases: base 10 (common logarithm), base e (natural logarithm, ln), base 2 (binary logarithm), and any custom base. The calculation uses the change of base formula: log_b(x) = ln(x) / ln(b).

Logarithms are essential in science (pH scale, Richter scale, decibels), computer science (binary search complexity, information theory), finance (continuous compounding), and engineering (signal processing).

When This Page Helps

Logarithms with arbitrary bases are not available on basic calculators. This calculator computes any log directly using the change-of-base formula.

How to Use the Inputs

Enter the number (x) to take the log of.
Enter the base (b) of the logarithm.
The result is log_b(x) = ln(x) / ln(b).
Common presets: base 10, base e, base 2.
View both the log result and the antilog verification.

Formula used

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

Where:
- b = base (must be > 0 and ≠ 1)
- x = the argument (must be > 0)
- ln = natural logarithm (base e)

Example Calculation

Result: 3

log₁₀(1000) = 3 because 10³ = 1000. Using the formula: ln(1000)/ln(10) = 6.9078/2.3026 = 3.

Tips & Best Practices

log_b(1) = 0 for any base, because b⁰ = 1.
log_b(b) = 1 for any base.
log₂(n) tells you the number of bits needed to represent n values.
The decibel scale uses 10 × log₁₀(ratio).
pH = −log₁₀[H⁺] concentration.
Logarithms convert multiplication into addition.

Logarithmic Scales

Many real-world measurements use logarithmic scales because the quantities span many orders of magnitude. The Richter scale for earthquakes, decibels for sound, pH for acidity, and stellar magnitude for brightness all use logarithms.

Logarithms in Finance

Continuous compounding uses the natural logarithm: the time to double an investment at rate r is ln(2)/r. The Rule of 72 (72/r%) is a simplified version of this.

Properties of Logarithms

log(ab) = log(a) + log(b), log(a/b) = log(a) − log(b), and log(a^n) = n×log(a). These properties simplify complex calculations and form the basis of the slide rule.

Consistent practice with varied problems builds computational fluency and deepens conceptual understanding that transfers across many technical fields.

Sources & Methodology

Last updated: February 8, 2026

Frequently Asked Questions

A logarithm answers: "To what exponent must I raise the base to get this number?" log₁₀(100) = 2 because 10² = 100. It is the inverse of exponentiation.