Log Calculator (Logarithm)
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Results
log₁₀(1,000) = 3
1. log₁₀(1,000) asks: 10 to what power gives 1,000? 2. log₁₀(1,000) = 3 3. Check: 10^3 = 1,000
| Power k | 10ᵏ | log₁₀ of it |
|---|---|---|
| 2 | 100 | 2 |
| 3 | 1,000 | 3 |
| 4 | 10,000 | 4 |
| 5 | 100,000 | 5 |
| 6 | 1,000,000 | 6 |
How to use the log calculator
- Choose what to find: the logarithm y, the number x (the antilog), or the base b.
- Pick base 10, e or 2, or choose “Another base” and type it. The base must be positive and not 1.
- Enter the values you know. The answer, the logs in the other common bases and the working update as you type.
Logarithm formula
Change of base: log_b(x) = ln x ÷ ln b = log₁₀ x ÷ log₁₀ b
Antilog: x = bʸ · Base: b = x^(1/y)
A logarithm answers the question “what power do I raise the base to, to get this number?”. So log₁₀ 1000 = 3 because 10³ = 1000, and log₂ 8 = 3. The base b must be positive and not 1, and the number x must be greater than 0, because a positive base raised to any real power is always positive. Calculators usually only have log (base 10) and ln (base e ≈ 2.71828), so other bases use the change-of-base rule.
Worked example: log₃ 50
There is no log₃ key, so divide natural logs: ln 50 = 3.912023005 and ln 3 = 1.098612289, so log₃ 50 = 3.912023005 ÷ 1.098612289 = 3.560876795. That makes sense: 3³ = 27 and 3⁴ = 81, so the answer must lie between 3 and 4. Checking, 3 raised to 3.560876795 is 50.
Logs in science and money
- pH: pH = −log₁₀[H⁺]. A solution with a hydrogen-ion concentration of 3.2 × 10⁻⁴ mol/L has pH = 3.495.
- Doubling time: money growing by 7% a year doubles when 1.07ⁿ = 2, so n = log₁.₀₇ 2 = 10.24 years.
- Antilog: if log₁₀ x = 2.5, then x = 10^2.5 = 316.22777.
- Unknown base: if log_b 1000 = 1.5, then b = 1000^(1/1.5) = 100.
Common logarithm values
| x | log₁₀ x | ln x | log₂ x |
|---|---|---|---|
| 0.1 | −1 | −2.30259 | −3.32193 |
| 0.5 | −0.30103 | −0.693147 | −1 |
| 1 | 0 | 0 | 0 |
| 2 | 0.30103 | 0.693147 | 1 |
| 5 | 0.69897 | 1.60944 | 2.32193 |
| 10 | 1 | 2.30259 | 3.32193 |
| 50 | 1.69897 | 3.91202 | 5.64386 |
| 100 | 2 | 4.60517 | 6.64386 |
| 1000 | 3 | 6.90776 | 9.96578 |
Laws of logarithms, checked
| Law | Example | Direct | Using the law |
|---|---|---|---|
| Product: log(ab) = log a + log b | log₁₀(20 × 50) | 3 | 3 |
| Quotient: log(a/b) = log a − log b | log₁₀(500 ÷ 5) | 2 | 2 |
| Power: log(aⁿ) = n log a | log₂(2¹⁰) | 10 | 10 |
| Change of base: log_b a = ln a ÷ ln b | log₃ 50 | 3.56088 | 3.56088 |
Tips and common mistakes
- log(a + b) is not log a + log b. The product law is about multiplication: log(ab) = log a + log b.
- Numbers between 0 and 1 have negative logs (in bases above 1): log₁₀ 0.1 = −1. A log of 0 or a negative number is undefined.
- Check which log your textbook means. “log” is base 10 in most school courses and on calculators, but often means ln in higher mathematics and base 2 in computing.
- Solving for an exponent, such as 2ⁿ = 1000, is the same as finding log₂ 1000; the exponent calculator does it directly. For mixed expressions use the scientific calculator.
Frequently asked questions
How do I calculate a log with a different base?
Use the change-of-base rule: log_b(x) = ln x ÷ ln b. For example, log₃ 50 = ln 50 ÷ ln 3 = 3.912023005 ÷ 1.098612289 ≈ 3.560876795.
What is the difference between log and ln?
log usually means the common logarithm, base 10, so log 100 = 2. ln is the natural logarithm, base e ≈ 2.71828, so ln e = 1. They differ by a constant factor: ln x ≈ 2.302585 × log₁₀ x.
Why is the log of a negative number undefined?
A logarithm is the power you raise a positive base to. A positive number raised to any real power is always positive, so no power gives 0 or a negative number.
What is an antilog?
The antilog undoes a logarithm: if log_b(x) = y, then x = bʸ. For base 10, the antilog of 2.5 is 10^2.5 ≈ 316.227766.
What is log base 2 used for?
log₂ counts doublings, so it appears in computing and information theory: log₂ 1024 = 10 because 2¹⁰ = 1024, which is why 10 bits can represent 1,024 different values.