log₂ 1024
- Calculate
- Log of x
- Base b
- 2
- Number x
- 1024
- Result
- 10
- Exact value
- 10
Checked against: 2¹⁰ = 1024
Logarithm calculator
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.
By AdminDrafted with AI writing toolsUpdated Checked examples: 6
Result
Result: 10
- Exact value
- 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
How it's calculated S
Change of base
Exact value
Check
About the logarithm calculator
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.
How to calculate a logarithm
A logarithm answers "what power of the base gives this number?": log₂ 8 = 3 because 2³ = 8. For any base, divide natural logarithms: log_b x = ln x ÷ ln b, so log₂ 10 = 2.302585 ÷ 0.693147 ≈ 3.321928. The base must be positive and not 1, and x must be positive.
Check which base a written "log" means. On most calculators and in school tables, log is base 10, while some mathematics texts and programming languages use log for base e. ln always means base e ≈ 2.718282, so log₁₀ 10 = 1 but ln 10 ≈ 2.302585.
Log table from 1 to 10
Common (base-10) and natural logarithms of 1 to 10, rounded to twelve decimal places:
| Number x | log10 x | ln x |
|---|---|---|
| 1 | 0 | 0 |
| 2 | 0.301029995664 | 0.69314718056 |
| 3 | 0.47712125472 | 1.098612288668 |
| 4 | 0.602059991328 | 1.38629436112 |
| 5 | 0.698970004336 | 1.609437912434 |
| 6 | 0.778151250384 | 1.791759469228 |
| 7 | 0.845098040014 | 1.945910149055 |
| 8 | 0.903089986992 | 2.07944154168 |
| 9 | 0.954242509439 | 2.197224577336 |
| 10 | 1 | 2.302585092994 |
Two rows hold in every base: log_b 1 = 0, because b⁰ = 1, and log_b b = 1. Powers of ten have whole-number common logs, so log₁₀ 100 = 2 and log₁₀ 1,000,000 = 6.
Other values follow from the table. log₁₀ 20 = log₁₀ 2 + log₁₀ 10 ≈ 1.30103, and log₁₀ 0.5 = −log₁₀ 2 ≈ −0.30103. For e itself, ln e = 1, log₁₀ e ≈ 0.434294481903 and log₂ e ≈ 1.442695040889.
Change of base and exponential equations
The change-of-base rule, log_b x = ln x ÷ ln b, is equation 4.2.8 in the NIST Digital Library of Mathematical Functions. For base 4 and x = 8 it gives ln 8 ÷ ln 4 = 1.5, which is exact because 4^(3/2) = (√4)³ = 8.
The same rule solves an exponential equation. For 3ˣ = 20, take logs of both sides: x ln 3 = ln 20, so x = ln 20 ÷ ln 3 ≈ 2.726833027861. The divisor is ln 3, not 3.
Logarithms can be negative. With a base between 0 and 1, the log of 8 to base 0.5 is −3, because 0.5⁻³ = 8. With a base above 1, any x below 1 has a negative log: log₁₀ 0.001 = −3, since 10⁻³ = 0.001.
Log rules: product, quotient and power
For positive a and c and one base b:
- Product: log_b(ac) = log_b a + log_b c.
- Quotient: log_b(a/c) = log_b a − log_b c.
- Power: log_b(aʳ) = r log_b a.
There is no rule for a sum: log_b(a + c) is not log_b a + log_b c. To expand log₁₀ 385, factor 385 = 5 × 7 × 11, so log₁₀ 385 = log₁₀ 5 + log₁₀ 7 + log₁₀ 11 ≈ 2.585460729509.
Write the base as a subscript or say it in words. "log ab" usually means the log of the product a × b, while log_a b is the logarithm of b to base a.
Why log 0 is undefined
Every real power of a positive base is positive, so no exponent gives 0 or a negative number. As x shrinks toward 0, log₁₀ x falls without bound: −1 at 0.1, −2 at 0.01, −3 at 0.001. It never reaches a value at 0 itself.
So a log table that starts at 0 needs an undefined entry there, and a tiny positive number rounded to 0 turns a valid problem into an invalid one.
Base 1 is excluded because 1 raised to any power is still 1, so it can never reach another number.
Where logarithms show up
Log scales compress huge ranges into small numbers. Sound intensity level is 10 × log₁₀ of the intensity ratio, so a sound 1,000 times the reference is 10 × 3 = 30 dB.
In chemistry, pH = −log₁₀ of the hydrogen-ion concentration in mol/L, so 10⁻⁷ mol/L gives pH 7.
Both scales fix the unit or reference before taking the log. Changing the unit changes the number inside the log, and with it the answer.
Entering a logarithm
- Type e as the base for a natural logarithm, or 10 for a common one.
- For an equation bˣ = y, the solve mode returns the exponent x directly.
- An exact value appears when the logarithm is a fraction, such as log₄ 8 = 3/2.
For a longer numerical expression use the scientific calculator, and to see the curve y = log_b x, plot it in the graphing calculator.
Worked examples
log₁₀ 0.001
- Calculate
- Log of x
- Base b
- 10
- Number x
- 0.001
- Result
- -3
- Exact value
- −3
Checked against: 10⁻³ = 0.001
ln 10
- Calculate
- Log of x
- Base b
- e
- Number x
- 10
- Result
- 2.302585092994
Checked against: Independent calculation
log₄ 8
- Calculate
- Log of x
- Base b
- 4
- Number x
- 8
- Result
- 1.5
- Exact value
- 3/2
Checked against: 4^(3/2) = (√4)³ = 8
Questions
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.
About this calculator
Sources
Checked against references
This calculator includes 6 worked examples with independently sourced answers. They run in the test suite, and you can run them here too.