Log Calculator: Common, Natural, Binary & Custom Base
Free log calculator: common log, natural log, log base 2, and custom base. Change of base formula steps, calculate ln(x) with steps. Logarithm and exponent inverse; characteristic and mantissa. No sign-upโall calculations run locally.
What This Calculator Does & Who It's For
Calculator Purpose & Ideal Users
- What You'll Get:Result: logb(x) = y. Inverse proof: Displays by = x. Change of base: For custom base, shows logb(x) = ln(x) / ln(b) with numeric steps. Domain validation: Explains when the number or base is invalid (x โค 0, base โค 0 or base = 1).
- Scope & Limits:Positive numbers only (x > 0). Base must be positive and not equal to 1. Results within JavaScript floating-point range. All calculations run in your browser; no data is sent to servers.
This log calculator computes logarithms for base 10 (common log), base e (natural log), base 2 (binary log), and any custom base. It shows the result (up to 10 decimal places), the inverse proof (by = x), and for custom base the change of base formula steps using ln(x) / ln(b).
What Is a Logarithm? Inverse of Exponentiation
A logarithm answers: "To what power must we raise the base to get this number?" So
means
For example, logโโ(100) = 2 because 10ยฒ = 100. The common log uses base 10 (written log or logโโ); the natural log uses base e (โ 2.718), written ln. Logarithms and exponents are inverses: logb(by) = y and blog_b(x) = x. For "What is log base e?" and "Why is log(1) always 0?", see the FAQ above
Common Log, Natural Log, Binary Log, Custom Base
Common log (base 10): Used in pH scales (pH = โlogโโ[Hโบ]), decibel measurements, and orders of magnitude. Natural log (base e): Essential for growth/decay (e.g. continuous compounding) and calculus (derivative of ln(x) is 1/x). Use the calculator to calculate ln(x) with steps for any positive x. Binary log (base 2), or log base 2: Primary in computer scienceโbits, information theory, and algorithms. Custom base: Use the change of base formula: logb(x) = ln(x) / ln(b). The calculator shows these steps when you select "Custom base."
Characteristic and Mantissa
In decimal form, a logarithm can be split into characteristic (integer part) and mantissa (fractional part). For example, logโโ(150) โ 2.176: the characteristic is 2 (so 10ยฒ โค 150 < 10ยณ) and the mantissa is 0.176. Historically, log tables (e.g. John Napier's) listed mantissas; you found the characteristic from the size of the number. This calculator gives the full decimal result; the integer part is the characteristic and the remainder is the mantissa.
John Napier and Log Tables
John Napier (1550โ1617) published the first tables of logarithms, which turned multiplication into addition and division into subtractionโenormously simplifying calculation before electronic computers. Napier's work used a base related to 1/e; later common logarithms (base 10) became standard for hand calculation. Today we use natural logarithms (base e) in calculus and science. This log calculator gives instant results for any base; the article explains the change of base formula so you can relate different bases.
Three Fundamental Log Laws: Product, Quotient, Power
Product rule:
โthe log of a product is the sum of the logs. Quotient rule:
โthe log of a quotient is the difference of the logs. Power rule:
โthe log of a power brings the exponent down. These follow from the definition by = x and the laws of exponents. Use this calculator to compute single logarithms; the inverse proof box reinforces the relationship b^y = x
Change of Base Formula
To compute log_b(x) using only natural or common log:
So you can find any logarithm with a calculator that has ln or log. Select "Custom base" in this log calculator to see the change of base formula steps applied to your base and number. For a quick summary, see the FAQ "What is the change of base formula?" above
Log Calculator FAQ
How do I use the log base calculator for a custom base?
Select "Custom base" from the mode selector. Enter your base (e.g. 5) in the base field and your number (e.g. 125) in the number field. The log base calculator will compute the result and display the change of base formula steps: logb(x) = ln(x) / ln(b).
What does "log base" mean?
The base of a logarithm is the number being raised to a power. In logโ
(125) = 3, the base is 5 โ meaning 5ยณ = 125. Common bases are 10 (common log), e (natural log), and 2 (binary log), but this log base calculator accepts any positive base other than 1.
What is log base e?
Log base e is the natural logarithm, written ln(x). The base e (Euler's number, โ 2.718) is the natural choice for growth, decay, and calculus. So ln(x) = loge(x). For example, ln(e) = 1 because eยน = e. Select "Natural log (base e)" in the calculator to compute ln(x).
Can you have a negative log base?
No. Logarithms are defined only for a positive base not equal to 1. If the base were negative or zero, the exponential function by would not be well-behaved for all real y (e.g. negative base with fractional exponent gives non-real results). Base 1 is excluded because 1y = 1 for every y, so there is no unique exponent. This calculator requires base > 0 and base โ 1.
Why is log(1) always 0?
Because bโฐ = 1 for any positive base b โ 1. The logarithm asks: "To what power must we raise the base to get this number?" So logb(1) = 0 means bโฐ = 1. That holds by definition of zero exponent. So logโโ(1) = 0, ln(1) = 0, and logb(1) = 0 for any valid base.
How do you convert an exponential equation to a logarithmic one?
If by = x, then logb(x) = y. The base b stays the base, the result x becomes the argument of the log, and the exponent y becomes the value of the log. Example: 10ยฒ = 100 โ logโโ(100) = 2. The calculator's "Inverse proof" box shows this relationship for your result.
What is the change of base formula?
To evaluate log_b(x) using another base (e.g. 10 or e):
So you can compute any log with a calculator that only has ln or logโโ. Select "Custom base" in this calculator to see the change of base formula steps