CalcOpenly
اردو

لوگارتھم کیلکولیٹر

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.

تازہ کاری جانچی گئی مثالیں: 6

Type e for the natural logarithm
آزمائیں
نتیجہ
نتیجہ: 10
زیادہ سے زیادہ اعشاری مقامات: 12؛ قریب ترین؛ برابر فاصلے پر صفر سے دور
عین قدر
10
Natural log of the number
6.931471805599
Base-10 log of the number
3.01029995664

log base 2 of 1024 is 10: 2 must be raised to the power 10 to give 1024.

y = log base 2 of x

05105001,0001,5002,000xy(1024, 10)
حساب کا طریقہ S

  1. Change of base

    log⁡21,024=ln⁡1,024ln⁡2=6.931471805599450.693147180559945=10\log_{2} 1{,}024 = \frac{\ln 1{,}024}{\ln 2} = \frac{6.93147180559945}{0.693147180559945} = 10

  2. عین قدر

    210=1,024  ⇒  log⁡21,024=102^{10} = 1{,}024 \;\Rightarrow\; \log_{2} 1{,}024 = 10

  3. Check

    210=1,0242^{10} = 1{,}024

لوگارتھم کیلکولیٹر کے بارے میں

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.

حل شدہ مثالیں

log₂ 1024

Calculate
Log of x
Base b
2
Number x
1024
نتیجہ
10
عین قدر
10

جانچ کا ماخذ: ⁨2¹⁰ = 1024⁩

log₁₀ 0.001

Calculate
Log of x
Base b
10
Number x
0.001
نتیجہ
-3
عین قدر
−3

جانچ کا ماخذ: ⁨10⁻³ = 0.001⁩

ln 10

Calculate
Log of x
Base b
e
Number x
10
نتیجہ
2.302585092994

جانچ کا ماخذ: ⁨Python Decimal(10).ln() = 2.302585092994045684017991454684364…⁩

log₄ 8

Calculate
Log of x
Base b
4
Number x
8
نتیجہ
1.5
عین قدر
3/2

جانچ کا ماخذ: ⁨4^(3/2) = (√4)³ = 8⁩

سوالات

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.

“⁨لوگارتھم کیلکولیٹر⁩” کتنا درست ہے؟

درستی آپ کی درج کردہ قدروں اور طریقے کے مفروضوں پر منحصر ہے۔ اعشاری حساب 50 بامعنی ہندسے استعمال کرتا ہے، مگر تخمینے، عددی طریقے اور ماخذ کا ڈیٹا کم درست ہو سکتے ہیں؛ دکھائی گئی قدروں کو راؤنڈ کرنے سے یہ حدود ختم نہیں ہوتیں۔ آزاد ذرائع کی حل شدہ مثالوں سے جانچ: 6۔ مثلاً، “⁨log₂ 1024⁩” کو ⁨2¹⁰ = 1024⁩ سے جانچا جاتا ہے۔

اس طریقے کا ماخذ کیا ہے؟

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

اس کیلکولیٹر کے بارے میں

log⁡bx=ln⁡xln⁡bbx=y  ⟺  x=ln⁡yln⁡b\begin{gathered} \log_b x = \frac{\ln x}{\ln b} \\[10pt] b^{x} = y \iff x = \frac{\ln y}{\ln b} \end{gathered}

ماخذ

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

حوالوں سے جانچ شدہ

اس کیلکولیٹر میں آزاد ذرائع سے حاصل کردہ جوابات والی 6 حل شدہ مثالیں شامل ہیں۔ یہ ٹیسٹ مجموعے میں چلتی ہیں اور آپ انہیں یہاں بھی چلا سکتے ہیں۔

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