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

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: \(t=\ln(F/P)/\ln(1+r)\).