Data Source and Methodology
Authoritative reference: NIST Digital Library of Mathematical Functions (DLMF), Chapter 4 “Elementary Functions”, §4.1–4.2 (Logarithms and Exponentials), 2024. Direct link: https://dlmf.nist.gov/4.
Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.
The Formula Explained
Given valid x > 0 and b > 0, b \neq 1:
1) \; y = \log_b(x) = \frac{\ln(x)}{\ln(b)}
2) \; x = b^{\,y}
3) \; b = x^{\frac{1}{y}} \quad (y \neq 0)
4) \; \log_b(x) = \frac{\log_{10}(x)}{\log_{10}(b)} = \frac{\ln(x)}{\ln(b)}
Glossary of Variables
- x (argument): The positive number whose logarithm is taken (x > 0).
- b (base): The logarithm base; must satisfy b > 0 and b ≠ 1. Common choices: e (natural), 10 (common), 2 (binary).
- y (result): The exponent such that by = x.
- ln(x): Natural logarithm with base e.
- log10(x): Common logarithm with base 10.
- Antilog: Solving x = by, the inverse of the logarithm.
- Precision: Reported either in decimal places (dp) or significant figures (sf).
Worked Example
How It Works: A Step-by-Step Example
Find y = log3(81).
- Identify variables: x = 81, b = 3, unknown y.
- Use the change-of-base formula: y = ln(x) / ln(b).
- Compute: ln(81) = 4.394449..., ln(3) = 1.098612....
- Divide: y = 4.394449... / 1.098612... = 4.
- Verify: 34 = 81. ✓
Frequently Asked Questions (FAQ)
What is a logarithm?
A logarithm answers the question “what power gives x from base b?” Formally, y = logb(x) means by = x.
Why must x be positive and b be positive but not 1?
From the definition by = x: for real logs, b must be positive and not 1, and x must be positive. These are standard domain restrictions.
What is the difference between ln and log?
ln denotes the natural logarithm (base e ≈ 2.71828). log without a base often means log10 (common) in engineering or ln in higher math—this tool labels bases explicitly to avoid ambiguity.
How do I compute an antilog?
Use x = by. In the calculator, switch to “Solve for x (antilog)”.
Can I solve for the base?
Yes. Given x and y, compute b = x1/y when y ≠ 0. If x = 1 and y ≠ 0, there is no valid base b > 0, b ≠ 1.
Do you support scientific notation and constants?
Yes, enter numbers like 3.5e-7. You can also use e and pi for the base or argument.
How do precision settings work?
Choose decimal places (dp) or significant figures (sf) and set a digit count (0–12). The tool computes with full precision and formats outputs per your selection.