# Logarithm and Other Important Properties in Algebra

**Properties of Logarithm**

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

**Other Important Properties in Algebra**

- $x \times 0 = 0$
- If $xy = 0$, then either
*x*= 0 or*y*= 0 or both*x*and*y*are zero - $\dfrac{0}{x} = 0$, provided
*x*≠ 0 - $\dfrac{x}{\infty} = 0$
- $\dfrac{0}{0} = \infty$
- $\dfrac{x}{0} = \infty$
- $0^0 = \infty$
- $1^\infty = \infty$
- $\infty^0 = \infty$
- $\infty - \infty = \infty$
- $0 \times \infty = \infty$
- $\dfrac{0}{\infty} = 0$
- $w^\infty = 0$
- $z^{-\infty} = 0$
- $0^x = 0$
- $0 \times x = 0$

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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