Logarithms formulas

Logarithms formulas

Positive real numbers: x, y, a, c, k

Natural number: n

1. Definition of logarithm

y = $log _{a}$ x if and only if x = a^y , a > 0 , a ≠ 1.

2. $log _{a} 1$ = 0

3. $log _{a} 1$ = 1

4. $log _{a} 0 = \left\{\begin{matrix} -\infty & if a > 1\\ +\infty & if a < 1 \end{matrix}\right.$

5. $log _{a} (xy) = log _{a} x + log _{a}y$

6. $log _{a} (\frac{x}{y}) = log _{a} x - log _{a}y$

7. $log _{a} (x^{n}) = n log _{a} x$

8. $log _{a} (\sqrt[n]{x}) = \frac{1 }{n}log _{a} x$

9. $log _{a}x = \frac{log _{c}x}{log _{c}a }= log _{c}x\cdot log _{a}c , c > 0 , c \neq 1.$

10. $\log _{a}c = \frac{1}{\log _{c}a}$

11. $x = a^{\log _{a}x}$

12. Logarithm to Base 10

$log_{10}x = log x$

13. Natural Logarithm

$log_{e}x = ln x$ ,

Where $e = \lim_{k\rightarrow \infty }(1+\frac{1}{k})^{k} = 2.718281828$

14. $log_{x}=\frac{1}{ln10}lnx = 0.434294 lnx$

15. $lnx=\frac{1}{log e}log x = 2.302585 logx$