# Logarithms

• Logarithm of a number $$b$$ to base $$a:$$ $${\log_a}b$$
Logarithm bases: $$a$$, $$d$$
Logarithm value (exponent): $$x$$
Real numbers: $$p$$, $$q$$
Positive real numbers: $$a$$, $$b$$, $$c$$, $$d$$, $$x$$, $$y$$
1. 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 $$b \gt 0,$$ $$a \gt 0,$$ $$a \ne 1$$
2. Logarithm of one $${\log_a}1 = 0$$
3. Logarithm of a number to the same base $${\log_a}a = 1$$
4. Logarithm of a product $${\log_a}{(bc)} = {\log_a}b + {\log_a}c$$
5. Logarithm of a quotient $${\log_a}{(b/c)} = {\log_a}b – {\log_a}c$$
6. Logarithm of a power $${\log _a}\left( {{b^p}} \right) = p\,{\log _a}b$$
7. Logarithm of a root $${\log _a}\sqrt[\large p\normalsize]{b} = {\large\frac{1}{p}\normalsize}{\log _a}b$$
8. $${\log _{{a^{\large q\normalsize}}}}b = {\large\frac{1}{q}\normalsize}{\log _a}b$$
9. $${\log _{{a^{\large q\normalsize}}}}{b^p} = {\large\frac{p}{q}\normalsize}{\log _a}b$$
10. Logarithm base change
$${\log _a}b = {\large\frac{{{{\log }_d}b}}{{{{\log }_d}a}}\normalsize},$$ where $$d \ne 1.$$
11. $${\log _a}b = {\large\frac{1}{{{{\log }_b}a}}\normalsize},$$ where $$b \ne 1.$$
12. Cancelling exponentials
$${a^{\large{{\log }_a}b}\normalsize} = b,$$ where $$b \gt 0,$$ $$a \gt 0,$$ $$a \ne 1.$$
13. The common logarithm is called the logarithm to the base $$10.$$ It is denoted as $${\log _{10}}b = \log b = \lg b.$$
14. 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 $${\log _e}b = \ln b.$$
15. The number $$e$$ as a limit $$e = \lim\limits_{x \to \infty } \left( {1 + \large\frac{1}{x}}\normalsize \right)$$
16. Conversion factor from natural to common logarithms
$$M = 1/\ln 10 =$$ $$\lg e \approx 0.4343 \ldots$$
17. Converting natural logarithms to common logarithms
$$\lg b = M \cdot \ln b \approx$$ $$0.4343\ln b$$
18. Converting common logarithms to natural logarithms
$$\ln a = 1/M \cdot \lg b \approx$$ $$2.3026\lg b$$