Logarithm & Antilog Calculator
log, ln and log_b to high precision — with change of base, log laws and log tables.
Logarithm
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Change of base
Log-table form
Antilogarithm
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Solution
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Result
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About the Logarithm & Antilog Calculator
Type a number to get its logarithm in any base — log₁₀, ln, log₂ or a base you choose — to as many as 500 significant digits, with ln x, log₁₀ x and log₂ x side by side. When the answer is a fraction it is recognised and shown exactly: log₄ 8 = 3/2, because 4^(3/2) = 8. The change-of-base formula is worked out, and the log-table form — characteristic, four-figure mantissa and bar notation such as 2̄.3692 — is explained for anyone using printed tables.
The other modes find antilogs (10ʸ, eʸ, 2ʸ or bʸ, including antilogs of bar notation), solve log_b(x) = y and bˣ = y for whichever value is unknown, and expand or condense expressions with the product, quotient and power rules, showing each law as it is used.
How to use it
- Choose Logarithm, Antilog, Solve or Log laws.
- Type the number. Fractions (
1/8), powers (2^10,e^3), roots (√2),πand E notation (6.02e23) all work. - Pick the base — log₁₀, ln, log₂, or log_b with a base of your own — and the number of significant digits.
- For log laws, type an expression such as
log(x^2 y / z^3)to expand, or2 log(x) + log(y) − 3 log(z)to condense. - Read the answer and the steps, and copy the result.
Examples
log₁₀ 1000
3
Because 10³ = 1000.
ln 2
≈ 0.693147180559945
log₄ 8
3/2 = 1.5
4^(3/2) = (√4)³ = 8.
log₃ 20
≈ 2.72683302786084
ln 20 ÷ ln 3 = 2.995732… ÷ 1.098612….
log 0.0234
2̄.3692 = −1.6308
Characteristic −2 (bar 2); the mantissa .3692 is the table entry at row 23, column 4.
antilog 2.3010
≈ 199.986
10^0.3010 ≈ 1.99986, times 10².
3ˣ = 20
x = log₃ 20 ≈ 2.7268
log(x²y/z³)
2 log(x) + log(y) − 3 log(z)
1/2 ln(x) − ln(y)
ln(√x/y)
Common uses
- Homework on logarithms, exponential equations and the log laws.
- Working with printed log tables, or checking answers found with them.
- Getting high-precision values of logarithms for science and engineering work.
Definitions
The logarithm to base b of x is the power you raise b to in order to get x: log_b x = y means bʸ = x. It is defined for x > 0 and for a base b > 0 with b ≠ 1. The natural logarithm ln x uses base e ≈ 2.718281828, the common logarithm log x uses base 10, and the binary logarithm log₂ x uses base 2. These follow the definitions in the NIST Digital Library of Mathematical Functions (§4.2): ln is the inverse of the exponential function, and log_b x = ln x / ln b.
The laws of logarithms
- Product rule: log_b(MN) = log_b M + log_b N
- Quotient rule: log_b(M/N) = log_b M − log_b N
- Power rule: log_b(Mᵖ) = p · log_b M — roots are powers too, so log_b √M = ½ log_b M
- Change of base: log_b x = log_k x ÷ log_k b for any base k, usually 10 or e
- Special values: log_b 1 = 0 and log_b b = 1
There is no rule for the logarithm of a sum: log(x + y) is not log x + log y. The laws assume the variables are positive, so that every logarithm is defined.
Characteristic and mantissa (log tables)
Before calculators, logarithms were looked up in four-figure tables. Write the number in scientific notation, a × 10ⁿ with 1 ≤ a < 10. The characteristic is n; the mantissa is log₁₀ a, a value from 0 to 1 read from the table. For 0.0234 = 2.34 × 10⁻², the characteristic is −2 and the table gives 0.3692 for 2.34, so log 0.0234 = −2 + 0.3692. Tables keep the mantissa positive and write a negative characteristic with a bar: 2̄.3692, which equals −1.6308. To find an antilog, reverse the process: 10^(2̄.3692) = 10^0.3692 × 10⁻² ≈ 2.340 × 0.01 = 0.02340.
Accuracy
Values are computed with decimal arithmetic at the chosen precision plus guard digits (using the decimal.js library) and rounded to the number of significant digits you choose, up to 500. Exact answers are confirmed with whole-number arithmetic — for example 4³ = 8², so log₄ 8 = 3/2 exactly. The four-figure mantissa is rounded half up, as in printed tables.
Limitations
- Only real logarithms of positive numbers are calculated. For a negative number the tool gives the principal complex value of ln instead, ln|x| + πi.
- Precision goes up to 500 significant digits; the highest settings take a moment longer.
- Log laws handle logarithms of products, quotients, powers and roots. Logarithms of sums, logarithms inside logarithms and mixed bases in one condense are not combined — change the base first.
- In log-law expressions each letter is its own variable (xy means x times y); e is Euler’s number and π is pi.
Privacy
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Frequently asked questions
What is the difference between log and ln?
log usually means the common logarithm, base 10, and ln the natural logarithm, base e ≈ 2.71828. They differ by a constant factor: ln x = ln 10 × log x ≈ 2.302585 × log x. In many programming languages, though, log means the natural logarithm.
How do I calculate a logarithm with a different base?
Use the change-of-base formula: log_b x = ln x ÷ ln b (or log x ÷ log b). For example, log₃ 20 = ln 20 ÷ ln 3 ≈ 2.9957 ÷ 1.0986 ≈ 2.7268.
What is an antilog?
The inverse of a logarithm: the antilog of y to base b is bʸ. The common antilog of 2.3010 is 10^2.3010 ≈ 200, because log 200 ≈ 2.3010.
Why is the log of a negative number undefined?
A positive base raised to any real power is positive, so no real power gives a negative number or zero. In the complex numbers, ln(−x) = ln x + πi is the principal value.
How do I solve 3ˣ = 20?
Take logarithms of both sides: x = log₃ 20 = ln 20 ÷ ln 3 ≈ 2.7268. Check: 3^2.7268 ≈ 20.
What are the characteristic and mantissa of a logarithm?
The characteristic is the whole-number part and the mantissa the decimal part, chosen so that the mantissa is between 0 and 1. For log 5000 = 3.6990, the characteristic is 3 (one less than the number of digits before the decimal point) and the mantissa is 0.6990, the table value for 5.000.
How do I expand log(x²y/z³)?
Apply the quotient rule, then the product rule, then the power rule: log(x²y/z³) = log(x²y) − log(z³) = log x² + log y − log z³ = 2 log x + log y − 3 log z.