# Logarithmusrechner

> The logarithm of a number to any base, including ln and log₁₀, with change-of-base steps, or the exponent x that solves bˣ = y, exact when rational.

Interaktive Version: https://www.calcopenly.com/de/math/logarithm-calculator
Thema: Mathematikrechner

The logarithm log_b x is the exponent that turns the base b into x: log₂ 1024 = 10 because 2¹⁰ = 1024. Any base can be computed from natural logarithms with the change-of-base rule, log_b x = ln x ÷ ln b, and the same rule solves an exponential equation bˣ = y, giving x = ln y ÷ ln b.

Common uses are counting doublings or halvings (base 2), orders of magnitude and the decibel and pH scales (base 10), and continuous growth and decay (base e ≈ 2.71828). Solving 3ˣ = 20 gives x = ln 20 ÷ ln 3 ≈ 2.7268.

The number must be positive, and the base positive and not 1. When the answer is a fraction it is also shown exactly, so log₄ 8 appears as 3/2 as well as 1.5. A base below 1 gives negative logarithms for numbers above 1: log₀.₅ 8 = −3.

## Eingaben

- **Calculate** (Optionen: Log of x, Solve bˣ = y)
- **Base b**: Type e for the natural logarithm
- **Number x**
- **Value y**

## Ergebnisse

- Ergebnis — Hauptergebnis
- Exakter Wert
- Natural log of the number
- Base-10 log of the number

## Formel

$$
\begin{gathered} \log_b x = \frac{\ln x}{\ln b} \\[10pt] b^{x} = y \iff x = \frac{\ln y}{\ln b} \end{gathered}
$$

## Durchgerechnete Beispiele

### log₂ 1024

- Calculate: Log of x
- Base b: 2
- Number x: 1024
- **Ergebnis: 10**
- **Exakter Wert: 10**
- Prüfquelle: 2¹⁰ = 1024

### log₁₀ 0.001

- Calculate: Log of x
- Base b: 10
- Number x: 0.001
- **Ergebnis: -3**
- **Exakter Wert: −3**
- Prüfquelle: 10⁻³ = 0.001

### ln 10

- Calculate: Log of x
- Base b: e
- Number x: 10
- **Ergebnis: 2.302585092994**
- Prüfquelle: Python Decimal(10).ln() = 2.302585092994045684017991454684364…

### log₄ 8

- Calculate: Log of x
- Base b: 4
- Number x: 8
- **Ergebnis: 1.5**
- **Exakter Wert: 3/2**
- Prüfquelle: 4^(3/2) = (√4)³ = 8

### Solve 3ˣ = 20

- Calculate: Solve bˣ = y
- Base b: 3
- Value y: 20
- **Ergebnis: 2.726833027861**
- Prüfquelle: Python Decimal(20).ln() / Decimal(3).ln() at 60 digits = 2.72683302786084204139609463636416…

### Base below 1: log₀.₅ 8

- Calculate: Log of x
- Base b: 0.5
- Number x: 8
- **Ergebnis: -3**
- **Exakter Wert: −3**
- Prüfquelle: 0.5⁻³ = 2³ = 8

## Fragen

### What is the difference between log and ln?

ln is the natural logarithm, with base e ≈ 2.718282, while log on most calculators means base 10. Some textbooks and programming languages use log for the natural logarithm instead; Python's math.log(10) returns 2.302585. The two differ by a constant factor, ln x = ln 10 × log₁₀ x ≈ 2.302585 × log₁₀ x, so ln 10 ≈ 2.302585 while log₁₀ 10 = 1.

### How do you calculate a logarithm with a different base?

Use the change-of-base rule: log_b x = ln x ÷ ln b, or equally log₁₀ x ÷ log₁₀ b. For log₂ 1024 that is 6.931472 ÷ 0.693147 = 10. For log₄ 8 it gives 1.5, which is exact because 4^(3/2) = (√4)³ = 8. Any base works except 1, and the base must be positive.

### How do you solve an exponential equation like 3ˣ = 20?

Take logarithms of both sides: x × ln 3 = ln 20, so x = ln 20 ÷ ln 3 ≈ 2.995732 ÷ 1.098612 ≈ 2.726833. The same method finds doubling times: money growing 7% a year doubles when 1.07ˣ = 2, at x = ln 2 ÷ ln 1.07 ≈ 10.24 years.

### Why is the logarithm of 0 or a negative number undefined?

No real power of a positive base gives 0 or a negative number: 2ˣ is positive for every real x and only approaches 0 as x heads toward −∞. So log₂ 0 has no value and ln(−1) has no real value; in complex numbers ln(−1) = iπ. A base of 1 is excluded too, since 1ˣ = 1 for every x.

### What is log base 2 used for?

log₂ x counts how many times 1 must be doubled to reach x. log₂ 1024 = 10, so 1,024 = 2¹⁰ and whole numbers from 0 to 1,023 fit in 10 bits. It also bounds halving processes: a binary search of 1,024 sorted items needs at most ⌊log₂ 1024⌋ + 1 = 11 comparisons.

### Wie genau arbeitet „Logarithmusrechner“?

Die Genauigkeit hängt von Ihren Eingaben und den Annahmen der Methode ab. Die Dezimalrechnung nutzt 50 signifikante Stellen, doch Schätzungen, numerische Verfahren und Quelldaten können ungenauer sein. Die angezeigte Rundung beseitigt diese Grenzen nicht. Anhand unabhängiger Quellen geprüfte Rechenbeispiele: 6. Beispielsweise wird „log₂ 1024“ anhand von 2¹⁰ = 1024 geprüft.

### Woher stammt die Methode?

NIST Digital Library of Mathematical Functions §4.2 — logarithms, change of base (4.2.E17); Khan Academy — Change of base formula for logarithms.

## Quellen

- [NIST Digital Library of Mathematical Functions §4.2 — logarithms, change of base (4.2.E17)](https://dlmf.nist.gov/4.2)
- [Khan Academy — Change of base formula for logarithms](https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs/x2ec2f6f830c9fb89:change-of-base/a/logarithm-change-of-base-rule-intro)