Logarithm Calculator — ln, log10, log2, and any base

Enter a positive number x. Choose a base or set a custom one. We’ll compute \(\ln(x)\), \(\log_{10}(x)\), \(\log_{2}(x)\), and \(\log_b(x)\) with antilog.

Inputs & Actions

Results

Chosen base:
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\(\log_b(x)\) (selected):
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Antilog \(b^{y}\) (check):
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\(\ln(x)\):
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\(\log_{10}(x)\):
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\(\log_{2}(x)\):
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Tip: You can paste numbers in E-notation like 3.2e-7.

What this calculator uses

  • Change of base: \(\log_b(x)=\dfrac{\ln(x)}{\ln(b)}\) (with \(x>0\), \(b>0\), \(b\neq 1\)).
  • Antilog: If \(y=\log_b(x)\) then \(x=b^y\).
  • Common logs: \(\log_{10}(x)=\ln(x)/\ln(10)\), \(\log_2(x)=\ln(x)/\ln(2)\).

Logarithms Explained (ln, log10, log2, and log base b)

A logarithm answers the question: “To what power must the base be raised to get \(x\)?” In symbols, \(\log_b(x)=y \iff b^y=x\). Our calculator supports natural log \(\ln(x)\) (base \(e\)), common log \(\log_{10}(x)\), binary log \(\log_{2}(x)\), and any base \(b\) via change-of-base.

Key Formulas

  • Change of base: \(\displaystyle \log_b(x)=\frac{\ln(x)}{\ln(b)}\)  (\(x>0,\;b>0,\;b\neq1\)).
  • Antilog (inverse): if \(y=\log_b(x)\) then \(x=b^{y}\).
  • Product / Quotient: \(\log_b(xy)=\log_b x+\log_b y,\;\; \log_b\!\left(\frac{x}{y}\right)=\log_b x-\log_b y\).
  • Power rule: \(\log_b(x^p)=p\,\log_b x\).
  • Common bases: \(\log_{10}x=\dfrac{\ln x}{\ln 10},\;\; \log_{2}x=\dfrac{\ln x}{\ln 2}\).

Domain & Quick Checks

  • For real numbers, you must have \(x>0\) and \(b>0,\;b\neq1\).
  • \(\log_b(1)=0\), \(\log_b(b)=1\), and \(\ln(e)=1\).
  • \(\log_b(0)\) is undefined (tends to \(-\infty\)), and \(\log_b(x<0)\) is not real.

Real-World Uses

  • pH: \(\mathrm{pH}=-\log_{10}[\mathrm{H}^+]\).
  • Decibels: \(L=10\log_{10}\!\left(\frac{P}{P_0}\right)\).
  • Information/bits: bits for \(N\) outcomes = \(\log_{2} N\).
  • Growth & half-life: \(t=\ln(x/x_0)/k\).
  • Compound interest: \(t=\ln(F/P)/\ln(1+r)\).
Precision tip (optional)

For \(x\) very close to 1, \(\ln(1+u)\) with \(u=x-1\) is more stable; many browsers expose Math.log1p(u).

5 Fun Facts about Logarithms

Logs turn multiplication into addition

That’s why slide rules worked: \(\log_b(xy)=\log_b x + \log_b y\). Adding distances gave products.

Analog magic

Base switches via change-of-base

\(\log_b(x)=\ln(x)/\ln(b)\). Any calculator with ln or log10 can compute any base with this one formula.

Base freedom

Decibels are logs

Sound levels are \(10\log_{10}(P/P_0)\). Add 10 dB and you multiply intensity by 10—a tidy logarithmic ladder.

Audio math

Bits are base‑2 logs

Information in bits for \(N\) outcomes is \(\log_{2} N\). Doubling outcomes adds exactly one bit of uncertainty.

Info theory

ln is an area

\(\ln(x)\) equals the area under \(1/t\) from 1 to \(x\). Logs connect algebra, geometry, and calculus in one sweep.

Calculus link

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