Logarithm of a number \(b\) to base \(a:\) \({\log_a}b\)

Logarithm bases: \(a\), \(d\)

Logarithm value (exponent): \(x\)

Logarithm bases: \(a\), \(d\)

Logarithm value (exponent): \(x\)

Real numbers: \(p\), \(q\)

Positive real numbers: \(a\), \(b\), \(c\), \(d\), \(x\), \(y\)

Positive real numbers: \(a\), \(b\), \(c\), \(d\), \(x\), \(y\)

- The logarithm of a number \(b\) (\(b \gt 0\)) to base \(a\) (\(a \gt 0\), \(a \ne 1\)) is called the exponent \(x\) to which the base \(a\) must be raised to obtain the number \(b\):

\({\log _a}b = x\; \Leftrightarrow \;{a^x} = b,\) where \(b \gt 0,\) \(a \gt 0,\) \(a \ne 1\) - Logarithm of one \({\log_a}1 = 0\)
- Logarithm of a number to the same base \({\log_a}a = 1\)
- Logarithm of a product \({\log_a}{(bc)} = {\log_a}b + {\log_a}c\)
- Logarithm of a quotient \({\log_a}{(b/c)} = {\log_a}b – {\log_a}c\)
- Logarithm of a power \({\log _a}\left( {{b^p}} \right) = p\,{\log _a}b\)
- Logarithm of a root \({\log _a}\sqrt[\large p\normalsize]{b} = {\large\frac{1}{p}\normalsize}{\log _a}b\)
- \({\log _{{a^{\large q\normalsize}}}}b = {\large\frac{1}{q}\normalsize}{\log _a}b\)
- \({\log _{{a^{\large q\normalsize}}}}{b^p} = {\large\frac{p}{q}\normalsize}{\log _a}b\)
- Logarithm base change

\({\log _a}b = {\large\frac{{{{\log }_d}b}}{{{{\log }_d}a}}\normalsize},\) where \(d \ne 1.\) - \({\log _a}b = {\large\frac{1}{{{{\log }_b}a}}\normalsize},\) where \(b \ne 1.\)
- Cancelling exponentials

\({a^{\large{{\log }_a}b}\normalsize} = b,\) where \(b \gt 0,\) \(a \gt 0,\) \(a \ne 1.\) - The common logarithm is called the logarithm to the base \(10.\) It is denoted as \({\log _{10}}b = \log b = \lg b.\)
- The natural logarithm is called the logarithm to the base \(e,\) where the transcendental number \(e\) approximately equals to \(e \approx 2.718281828 \ldots \) The natural logarithm is denoted as \({\log _e}b = \ln b.\)
- The number \(e\) as a limit \(e = \lim\limits_{x \to \infty } {\left( {1 + \large\frac{1}{x}}\normalsize \right)}^x\)
- Conversion factor from natural to common logarithms

\(M = 1/\ln 10 =\) \( \lg e \approx 0.4343 \ldots \) - Converting natural logarithms to common logarithms

\(\lg b = M \cdot \ln b \approx\) \( 0.4343\ln b\) - Converting common logarithms to natural logarithms

\(\ln a = 1/M \cdot \lg b \approx\) \( 2.3026\lg b\)