Logarithm Calculator
Logarithms to base 10, base e, base 2 or any base you choose, with the change-of-base working.
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Result
- log(1000)
- 3.0000000000
- log₁₀
- 3.0000000000
- ln (natural log)
- 6.9077552790
- log₂
- 9.9657842847
Because 10^3 = 1000
What power of 10 gives 1000?
log_b(x) = ln x ÷ ln b
ln 1000 = 6.907755
ln 10 = 2.302585
6.907755 ÷ 2.302585
log(1000) = 3.0000000000
Formula — written step by step
log₁₀(1000)…
log_b(x) = ln x ÷ ln b
Asking: 10 to what power gives 1000?
10¹ = 10, 10² = 100, 10³ = 1000
ln 1000 = 6.907755, ln 10 = 2.302585
6.907755 ÷ 2.302585 = 3
log₁₀(1000) = 3
b is the base and x the number. A logarithm answers the question "what power of the base gives this number" — it is the inverse of exponentiation.
A worked example
How many bits to store a number
- Values to distinguish
- 1,000,000
- Base
- 2
- Each bit doubles the number of values
- log₂(1,000,000) = ln 1,000,000 ÷ ln 2
- = 13.815511 ÷ 0.693147
- = 19.931569
20 bits — you must round up
Nineteen bits give 524,288 values, which is not enough; twenty give 1,048,576. Whenever a logarithm answers a "how many" question about whole units, the answer is the ceiling, not the rounded value.
How to use this calculator
- Enter a positive number. Logarithms of zero and negative numbers do not exist among the real numbers.
- Choose a base: 10 for orders of magnitude, e for natural growth and decay, 2 for anything involving doubling or bits.
- Read the result as the power the base must be raised to.
- The three common bases are shown together, so a value can be read in whichever scale suits the problem.
What a logarithm asks
A logarithm is the inverse of a power. Where 10³ = 1000 asks "what is ten cubed", log₁₀(1000) = 3 asks "ten to what power gives a thousand". Every statement about exponents has an equivalent statement about logarithms.
This makes logarithms the natural tool wherever quantities span many orders of magnitude. The decibel scale, the pH scale and earthquake magnitudes are all logarithmic, which is why a magnitude 7 earthquake releases roughly thirty times the energy of a magnitude 6 rather than merely a seventh more.
The three bases you will meet
Base 10 — the common logarithm — counts orders of magnitude, and is what people mean when they say a number "has six digits". Base 2 counts doublings, which makes it the natural language of computing: log₂ of a number of values is the bits required to store it.
Base e, the natural logarithm, is the one that appears in calculus and in every process of continuous growth or decay. It looks like an odd choice until you differentiate: the natural log is the only logarithm whose rate of change is exactly 1/x, which is why it turns up in compound interest, radioactive decay and population models alike.
Change of base
Most calculators offer only log₁₀ and ln, yet any base can be obtained from either using log_b(x) = ln x ÷ ln b. The choice of the intermediate base does not matter — dividing two base-10 logs gives the identical answer to dividing two natural logs, because the conversion factor cancels.
This calculator uses the natural-log form internally. It is also the reason the domain is restricted: the logarithm of zero would require a power of the base that produces nothing, and no finite power does. As x approaches zero the logarithm falls without limit, so the value is undefined rather than infinite.
Logarithm rules
| Rule | Form |
|---|---|
| Product | log(xy) = log x + log y |
| Quotient | log(x ÷ y) = log x − log y |
| Power | log(xⁿ) = n × log x |
| Base of itself | log_b(b) = 1 |
| Log of one | log_b(1) = 0 |
| Change of base | log_b(x) = ln x ÷ ln b |
The product rule is why logarithms were invented: it turns multiplication into addition, which is what made slide rules and log tables possible for three centuries.
Frequently asked questions
What is the difference between log and ln?
By convention log means base 10 and ln means base e, approximately 2.71828. In some fields — pure mathematics especially — log is written for the natural logarithm, so it is worth checking the context.
Why can I not take the log of a negative number?
Because no real power of a positive base produces a negative result. Raising 10 to any real power always gives a positive number, so there is no real answer.
What is an antilog?
The inverse operation: raising the base to the power of your value. The antilog base 10 of 3 is 10³ = 1000. Use the exponent calculator for this.
Why is log(1) always zero?
Because any base raised to the power of zero is one. Whatever the base, the power needed to produce 1 is 0.