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   $x \ne 0$.

5. $\dfrac{x}{\infty} = 0$

6. $\dfrac{0}{\infty} = 0$

7. $\dfrac{0}{0} = \infty$

8. $\dfrac{x}{0} = \infty$

9. $0^0 = \infty$

10. $1^\infty = \infty$

11. $\infty^0 = \infty$

12. $\infty - \infty = \infty$

13. $0 \times \infty = \infty$

14. $\dfrac{0}{\infty} = 0$

15. $w^\infty = 0$

16. $z^{-\infty} = 0$

17. $0^x = 0$

18. $0 \times x = 0$
 

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

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