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How to use
A logarithm answers an exponent question rather than an ordinary division question: which exponent applied to a base produces the given value? For example, 2 raised to 8 equals 256, so log base 2 of 256 is 8. Choose natural logarithm for base e, common logarithm for base 10, binary logarithm for base 2, or a custom positive base other than 1. The inverse or antilogarithm mode computes the power from a base and an exponent. These operations are useful in exponential-growth models, order-of-magnitude calculations and algebra, but only when you distinguish the input value from the logarithm base and respect the domain restrictions of real-number logarithms.
- Select ln, base-10 log, base-2 log or a custom base to match the original equation; these modes are not interchangeable.
- Enter a strictly positive x; for a custom base enter a positive number different from 1.
- Read the computed exponent and check the displayed inverse-power equality, allowing for ordinary floating-point rounding.
- To calculate an antilogarithm, switch to inverse mode and enter the base and exponent rather than a logarithm input value.
How it is calculated
Example
Work through three different modes. Base 2 with x = 256 returns log₂(256) = 8 because 2⁸ = 256. Base 10 with x = 0.001 returns −3 because 10⁻³ = 0.001; a negative logarithm can therefore be perfectly valid for a positive input below one. For the natural log, ln(e²) = 2, with e² approximately 7.38906. In antilog mode, entering base 10 and exponent −3 returns 0.001. That last result is not the numerical reciprocal of log(0.001): antilog means an exponential operation, not division by a logarithm.
Important notes
For real logarithms, x must be greater than zero; the base must also be positive and cannot equal 1. A fractional base between zero and one is allowed, but the logarithmic function then decreases with increasing x. JavaScript floating-point calculations can produce small rounding differences; the inverse-power display is a numerical check, not an exact symbolic proof. Very large inverse powers may exceed finite machine range, so do not treat overflow as a meaningful real-number result. Domain restrictions still apply when you solve an equation or transform an inequality using logs.
Worked examples and interpreting results
A logarithm asks for the exponent: to what power must the base be raised to obtain the number? Thus log base 10 of 1000 is 3, and log base 2 of 32 is 5.
| Case | Calculation | Result |
|---|---|---|
| Base ten | log₁₀(1000) = 3 ⇔ 10³ = 1000 | 3 |
| Base two | log₂(32) = 5 ⇔ 2⁵ = 32 | 5 |
| Logarithm of one | log₁₀(1) | 0 |
| Inverse log: base 10, exponent −2 | 10^−2 | 0.01 |
| Logarithm of 0.125 to base 0.5 | ln(0.125) ÷ ln(0.5) | 3 |
How to check the result
For real logarithms, the input must be positive and the base positive and unequal to one. ln uses base e and differs from base-10 log. Check by raising the base to the result to recover the input.
Logarithm calculator for base 10, base 2 and ln
A logarithm determines the exponent needed for a given positive base to produce a number. For example, log base 10 of 1,000 is 3 because 10 cubed equals 1,000. Natural logarithms use the base e, and other bases can be evaluated by change of base.
- Base 10 log of 1,000: log₁₀(1000) = 3
- Base 2 log of 32: log₂(32) = 5
- Natural logarithm of e²: ln(e²) = 2
Calculation caution
In the real number system, the argument must be greater than zero and the base must be positive but not equal to one. Logarithms of zero or a negative argument are undefined.
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Frequently asked questions
What is the difference between ln, log10 and log2?
They use bases e, 10 and 2 respectively. The same positive x can have different logarithm values in each mode.
Can a logarithm give a negative result?
Yes. For example, log base 10 of 0.001 is −3 even though the argument is strictly positive.
Does the inverse logarithm mean reciprocal?
No. Antilog computes base raised to an exponent. It is not one divided by a logarithm value.