Logarithm and Other Important Properties in Algebra

Tags:

common logarithm

natural logarithm

Naperian logarithm

logarithm of a product

logarithm of a quotient

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

Submitted by Romel Verterra on June 8, 2011 - 12:02am

15917 reads

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