Author: Starlight Tools math editors
Reviewer: Starlight Robotics calculation QA
Last reviewed: July 5, 2026
Accuracy note: Results use JavaScript Math.log, Math.exp, and change-of-base formulas in your browser.
Tip: You can paste numbers in E-notation like 3.2e-7. Leave exactly one input blank.
| x | ln(x) | log10(x) | log2(x) |
|---|---|---|---|
| 0.001 | -6.9078 | -3 | -9.9658 |
| 0.01 | -4.6052 | -2 | -6.6439 |
| 0.1 | -2.3026 | -1 | -3.3219 |
| 1 | 0 | 0 | 0 |
| 2 | 0.6931 | 0.3010 | 1 |
| 10 | 2.3026 | 1 | 3.3219 |
| 100 | 4.6052 | 2 | 6.6439 |
| 1000 | 6.9078 | 3 | 9.9658 |
Logarithms help you solve “exponent questions” quickly. In symbols, \(\log_b(x)=y\) means \(b^y=x\). This calculator makes that relationship practical by solving for the missing argument \(x\), base \(b\), or result \(y\).
Think of a base \(b\) as the “step size” and the logarithm as the number of steps needed to reach \(x\). Because logs are the inverse of exponentiation, they are perfect for undoing growth, comparing scales, or converting between powers. For real-number results, the input must be positive (\(x>0\)), and the base must be positive and not equal to 1. When \(x\) is between 0 and 1, the log is negative, which simply means you need a fraction of a step to reach that smaller value.
This tool uses the change-of-base idea \(\log_b(x)=\ln(x)/\ln(b)\), so any base works even if you only know \(\ln\) or \(\log_{10}\). It also verifies the answer by checking the exponential form \(b^y=x\).
Many measurements are based on a logarithmic scale because they span huge ranges. Sound levels in decibels compare power ratios with \(\log_{10}\), acidity is measured with pH (also a base‑10 log), and information theory uses \(\log_{2}\) to count bits needed for a number of outcomes. In finance and science, logarithms simplify exponential growth or decay so you can solve for time, rates, or multipliers. As a quick intuition, every 10× change in a base‑10 log shifts the value by exactly 1, which makes comparisons fast and meaningful.
For \(x\) very close to 1, \(\ln(1+u)\) with \(u=x-1\) is more stable; many browsers expose Math.log1p(u).
Use \(\log_2(x)=\ln(x)/\ln(2)\). For example, \(\log_2(64)=6\) because \(2^6=64\).
In many school and calculator contexts, log means base 10 and ln means base \(e\). Some advanced math contexts use log for natural log, so check the convention.
Rewrite the logarithm as \(x=b^y\). For example, \(\log_3(x)=4\) gives \(x=3^4=81\).
For a positive real base \(b\neq1\), \(b^y\) is always positive. It cannot produce a negative \(x\), so \(\log_b(x)\) is undefined for negative \(x\) in real numbers.
Antilog is the inverse operation. If \(y=\log_b(x)\), then \(x=b^y\).
For bases greater than 1, \(\log_b(x)\) is negative between 0 and 1. For bases between 0 and 1, the sign reverses.
For real numbers, \(x>0\), \(b>0\), and \(b\neq1\). The exponent result \(y\) can be any real number when the domain is valid.
Yes. The calculator runs locally in your browser; values are not uploaded.
That’s why slide rules worked: \(\log_b(xy)=\log_b x + \log_b y\). Adding distances gave products.
\(\log_b(x)=\ln(x)/\ln(b)\). Any calculator with ln or log10 can compute any base with this one formula.
Sound levels are \(10\log_{10}(P/P_0)\). Add 10 dB and you multiply intensity by 10—a tidy logarithmic ladder.
Information in bits for \(N\) outcomes is \(\log_{2} N\). Doubling outcomes adds exactly one bit of uncertainty.
\(\ln(x)\) equals the area under \(1/t\) from 1 to \(x\). Logs connect algebra, geometry, and calculus in one sweep.
