Logarithm formulas

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Logarithm

Variables

a, b, x, y

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Positive real numbers

1.

Definition of logarithm

$\log_ba=x \iff b^x=a$

2.

Log of 1

$\log_a 1=0$

3.

Log of same numbers

$\log_a a = 1$

4.

Product rule for logarithms

$\log_a {xy}=\log_a x + \log_a y$

5.

Quotient rule for logarithms

$\log_a \dfrac{x}{y} = \log_ax - \log_ay$

6.

Power rule for logarithms

$\log_ax^n=n\log_ax$

7.

Log of square root

$\log_a\sqrt[n]{x}=\dfrac{1}{n} \log_ax$

8.

Change of base

$\log_ba=\dfrac{\log_ax}{\log_ay}$

9.

Change of base

$\log_ax=\dfrac{1}{\log_xa}$

10.

$a^{\log_ax}=x$

11.

Base 10 logarithm

$\log_{10}x = \log x$

12.

Natural logarithm

$\log_e x = \ln x, e=2.718281...$

13.

Log to Ln

$\log x = \dfrac{\ln x}{\ln 10}$

14.

Ln to Log

$\ln x = \dfrac{\log x}{\log e}$

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