# సంవర్గమాన కాలిక్యులేటర్

> 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.

ఇంటరాక్టివ్ వెర్షన్: https://www.calcopenly.com/te/math/logarithm-calculator
విషయం: గణిత కాలిక్యులేటర్లు

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.

## ఇన్‌పుట్‌లు

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

## ఫలితాలు

- ఫలితం — ప్రధాన ఫలితం
- ఖచ్చితమైన విలువ
- Natural log of the number
- Base-10 log of the number

## సూత్రం

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

## పరిష్కరించిన ఉదాహరణలు

### log₂ 1024

- Calculate: Log of x
- Base b: 2
- Number x: 1024
- **ఫలితం: 10**
- **ఖచ్చితమైన విలువ: 10**
- తనిఖీ చేసిన మూలం: 2¹⁰ = 1024

### log₁₀ 0.001

- Calculate: Log of x
- Base b: 10
- Number x: 0.001
- **ఫలితం: -3**
- **ఖచ్చితమైన విలువ: −3**
- తనిఖీ చేసిన మూలం: 10⁻³ = 0.001

### ln 10

- Calculate: Log of x
- Base b: e
- Number x: 10
- **ఫలితం: 2.302585092994**
- తనిఖీ చేసిన మూలం: Python Decimal(10).ln() = 2.302585092994045684017991454684364…

### log₄ 8

- Calculate: Log of x
- Base b: 4
- Number x: 8
- **ఫలితం: 1.5**
- **ఖచ్చితమైన విలువ: 3/2**
- తనిఖీ చేసిన మూలం: 4^(3/2) = (√4)³ = 8

### Solve 3ˣ = 20

- Calculate: Solve bˣ = y
- Base b: 3
- Value y: 20
- **ఫలితం: 2.726833027861**
- తనిఖీ చేసిన మూలం: 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
- **ఫలితం: -3**
- **ఖచ్చితమైన విలువ: −3**
- తనిఖీ చేసిన మూలం: 0.5⁻³ = 2³ = 8

## ప్రశ్నలు

### 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.

### “సంవర్గమాన కాలిక్యులేటర్” ఎంత కచ్చితమైనది?

కచ్చితత్వం మీ ఇన్‌పుట్‌లు, పద్ధతిలోని ఊహలపై ఆధారపడి ఉంటుంది. దశాంశ గణన 50 సార్థక అంకెలను ఉపయోగిస్తుంది. అంచనాలు, సంఖ్యా పద్ధతులు, మూల డేటా తక్కువ కచ్చితత్వంతో ఉండవచ్చు; ప్రదర్శనలో విలువలను రౌండ్ చేయడం ఈ పరిమితులను తొలగించదు. స్వతంత్ర మూలాల పరిష్కారాలతో తనిఖీ చేసిన ఉదాహరణలు: 6. ఉదాహరణకు, “log₂ 1024”ను 2¹⁰ = 1024తో తనిఖీ చేశారు.

### ఈ పద్ధతికి మూలం ఏమిటి?

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

## మూలాలు

- [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)