Logarithm and Other Important Properties in Algebra

Properties of Logarithm

1. If   $y = a^x$,   then   $\log_a y = x$.   ← Definition of logarithm
2. $\log_a xy = \log_a x + \log_a y$
3. $\log_a \dfrac{x}{y} = \log_a x - \log_a y$
4. $\log_a x^n = n \log_a x$
5. $\log_a a = 1$
6. $\log_a 1 = 0$
7. $\log_{10} x = \log x$   ←   Common logarithm
8. $\log_e x = \ln x$   ←   Naperian or natural logarithm
9. $\log_y x = \dfrac{\log x}{\log y} = \dfrac{\ln x}{\ln y}$   ←   Change base rule
10. If   $\log_a x = \log_a y$,   then   $x = y$.
11. If   $\log_a x = y$,   then   $x = {\rm antilog}_a \, y$.

Other Important Properties in Algebra

1. $x \times 0 = 0$
2. If   $xy = 0$,   then either x = 0 or y = 0 or both x and y are zero
3. $\dfrac{0}{x} = 0$,   provided x ≠ 0
4. $\dfrac{x}{\infty} = 0$
5. $\dfrac{0}{0} = \infty$
6. $\dfrac{x}{0} = \infty$
7. $0^0 = \infty$
8. $1^\infty = \infty$
9. $\infty^0 = \infty$
10. $\infty - \infty = \infty$
11. $0 \times \infty = \infty$
12. $\dfrac{0}{\infty} = 0$
13. $w^\infty = 0$
14. $z^{-\infty} = 0$
15. $0^x = 0$
16. $0 \times x = 0$

Where
a, n, x, and y = any number not equal to zero (unless it is specified)
w = any number greater than zero but less than 1
z = any number greater than 1
∞ = infinity, undefined