Enter a number and get its logarithm in base 10, base e, base 2 and any base you choose, with the change of base step shown. Switch to antilog to go back.
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This log calculator is for students, engineers and anyone working with growth, sound levels, pH or computer science. Type a number and you get its common log, natural log, binary log and a log in any base, all at once, with the change of base formula worked out. The antilog mode reverses it and raises the base to a power.
What Does a Log Calculator Do?
A log calculator finds the power you must raise a base to in order to get a number. For example, log base 10 of 1,000 is 3, because 10³ = 1,000. For a base your calculator lacks, use change of base: log₂(50) = ln(50) ÷ ln(2) = 3.912023 ÷ 0.693147 = 5.643856.
What Is a Logarithm?
A logarithm is the exponent that a base must be raised to in order to produce a given number, so log_b(x) = y means b^y = x. Logs turn multiplication into addition, which is why John Napier introduced them in 1614 to speed up hand calculations in astronomy and navigation.
log_b(x) = ln(x) ÷ ln(b) (change of base)
antilog_b(y) = b^y
Three bases are used most. The common logarithm, log₁₀, is base 10 and is used for decibels, the pH scale and orders of magnitude. The natural logarithm, ln, uses Euler’s number e ≈ 2.718282 and appears in growth, decay and calculus. The binary logarithm, log₂, counts doublings and bits in computer science.
How Do You Use the Change of Base Formula?
You use the change of base formula by dividing the log of the number by the log of the base, using any log your calculator has: log_b(x) = ln(x) ÷ ln(b) = log₁₀(x) ÷ log₁₀(b). For log₃(81), ln(81) ÷ ln(3) = 4.394449 ÷ 1.098612 = 4, which checks out because 3⁴ = 81. The tool does this for every custom base and shows the division.
| x | log₁₀(x) | ln(x) | log₂(x) |
|---|---|---|---|
| 0.01 | −2 | −4.60517 | −6.643856 |
| 0.5 | −0.30103 | −0.693147 | −1 |
| 1 | 0 | 0 | 0 |
| 2 | 0.30103 | 0.693147 | 1 |
| e ≈ 2.718282 | 0.434294 | 1 | 1.442695 |
| 8 | 0.90309 | 2.079442 | 3 |
| 10 | 1 | 2.302585 | 3.321928 |
| 50 | 1.69897 | 3.912023 | 5.643856 |
| 100 | 2 | 4.60517 | 6.643856 |
| 1,000 | 3 | 6.907755 | 9.965784 |
| 1,024 | 3.0103 | 6.931472 | 10 |
What Is an Antilog?
An antilog is the inverse of a log: it raises the base to the given power, so antilog₁₀(2.5) = 10^2.5 ≈ 316.227766. Before electronic calculators, people looked up antilogs in printed tables to turn a log answer back into an ordinary number. Switch the tool to Antilog mode, pick a base and type the exponent. For powers with whole number bases and exponents, the exponent calculator gives exact results, and the scientific notation converter helps read very large or small answers.
Why Can You Not Take the Log of 0 or a Negative Number?
You cannot take the log of 0 or a negative number because no real power of a positive base gives 0 or a negative result. 10 raised to any power is always above 0, so log₁₀(−5) has no real answer and log₁₀(0) is undefined. Numbers between 0 and 1 have negative logs, for example log₁₀(0.01) = −2. The base also has to be positive and not 1, since 1 raised to any power is still 1.
Worked Example
A programmer wants to know how many times 50 items can be halved, which is log base 2 of 50. With 50 entered and base 2 selected, the tool shows log₂(50) = 5.643856. The change of base line reads ln(50) ÷ ln(2) = 3.912023 ÷ 0.693147 = 5.643856, and the boxes add log₁₀(50) = 1.69897 and ln(50) = 3.912023. So a binary search over 50 items needs at most 6 steps. Switching to Antilog with base 10 and an exponent of 2.5 gives 10^2.5 = 316.227766. To round results for a report, try the rounding calculator.
How Do You Use the Log Calculator?
Pick log or antilog
Choose Logarithm to find a log, or Antilog to raise a base to a power.
Type the number
Enter the number x, such as 50, 0.01 or 1e6. In antilog mode, enter the exponent.
Choose the base
Pick 10, e or 2, or choose Other base and type any positive base except 1.
Read the log and the working
The big result shows the answer; the boxes give log₁₀, ln, log₂ and your base, with change of base below.
Frequently Asked Questions
What is log base 10 of 1000?
log₁₀(1,000) = 3, because 10 × 10 × 10 = 1,000. Each extra zero adds 1 to the common log, so log₁₀(100) = 2 and log₁₀(1,000,000) = 6.
What is the difference between log and ln?
log usually means base 10 and ln always means base e, about 2.718282. They differ by a fixed factor: ln(x) = 2.302585 × log₁₀(x). So ln(10) = 2.302585 while log₁₀(10) = 1.
What is ln of 1?
ln(1) = 0, and the log of 1 is 0 in every base, because any positive base raised to the power 0 equals 1. This is why log scales start at 1, not at 0.
How do I calculate log base 2 on a normal calculator?
Divide ln(x) by ln(2), or log(x) by log(2). For 1,024, ln(1,024) ÷ ln(2) = 6.931472 ÷ 0.693147 = 10, since 2 to the power 10 is 1,024.
What is the antilog of 3?
The antilog of 3 in base 10 is 10³ = 1,000. In base e it is e³ ≈ 20.085537, and in base 2 it is 2³ = 8. Always check which base the problem uses before taking an antilog.
Why is the log of a number between 0 and 1 negative?
A positive base must be raised to a negative power to give a value below 1. For example, 10 to the power −2 is 1/100 = 0.01, so log₁₀(0.01) = −2 and log₂(0.5) = −1.
Sources and Further Reading
- Wikipedia: Logarithm
- Wolfram MathWorld: Logarithm
- NIST Digital Library of Mathematical Functions: 4.2 Logarithm
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Last reviewed: October 2026