Here are some fundamental algebraic rules and properties related to arithmetic operations:

Properties of Operations Link to heading

Commutative Property

  • Addition: $a + b = b + a$
  • Multiplication: $ab = ba$

Associative Property

  • Addition: $(a + b) + c = a + (b + c)$
  • Multiplication: $(ab)c = a (bc)$

Distributive Property

  • Left distributive property: $a (b + c) = a b + a c$
  • Right distributive property: $(a + b) c = a c + b c$

Special Elements Link to heading

Identity Elements

  • Addition: $a + 0 = a$
  • Multiplication: $a \cdot 1 = a$

Inverse Elements

  • Addition: $a + (-a) = 0$
  • Multiplication: $a \frac{1}{a} = 1$ for $a \neq 0$

Exponents Link to heading

Basic Rules

  • Product Rule: $a^{m} a^{n} = a^{m+n}$
  • Power of a Power: $(a^{m})^{n} = a^{mn}$
  • Power of a Product: $a^{n} b^{n} = (ab)^{n}$
  • Quotient Rule: $\frac{a^{m}}{a^{n}} = a^{m-n} \quad \text{for } a \neq 0$

Special Rules

  • Zero Exponent Rule: $a^{0} = 1 \quad \text{for } a \neq 0$
  • Negative Exponent Rule: $a^{-n} = \frac{1}{a^n} \quad \text{for } a \neq 0$
  • Negative Exponent Rule ($n=1$): $a^{-1} = \frac{1}{a} \quad \text{for } a \neq 0$

Logarithms Link to heading

Basic Rules

  • Definition:$\log_{b}(a) = c \iff b^{c} = a$
  • Product Rule: $\log_{b}(ac) = \log_{b}(a) + \log_{b}(c)$
  • Quotient Rule: $\log_{b}\left(\frac{a}{c}\right) = \log_{b}(a) - \log_{b}(c)$
  • Power Rule: $\log_{b}(a^{c}) = c \log_{b}(a)$
  • Change of Base Rule: $\log_{b}(a) = \frac{\log_{c}(a)}{\log_{c}(b)}$
  • Inverse Rule: $b^{\log_{b}(a)} = a$
$$\log_{b}(a) = c \iff b^{c} = a$$$$b^{\log_{b}(a)} = a$$

Natural Logarithm Link to heading

The natural logarithm is a logarithm to the base $e$, where $e$ is the Euler number. The natural logarithm is denoted by $\ln(x)$ which $\ln(x) = \log_{e}(x)$. We can apply all the logarithmic rules to the natural logarithm.

  • Definition: $\ln(a) = c \iff e^{c} = a$
  • Product Rule: $\ln(ac) = \ln(a) + \ln(c)$
  • Quotient Rule: $\ln\left(\frac{a}{c}\right) = \ln(a) - \ln(c)$
  • Power Rule: $\ln(a^{c}) = c \ln(a)$
  • Change of Base Rule: $\ln(a) = \frac{\log_{b}(a)}{\log_{b}(e)} = \frac{\log_{b}(a)}{\frac{1}{\ln(b)}} = \ln(b) \log_{b}(a)$
  • Inverse Rule: $e^{\ln(a)} = a$
  • Inverse Rule (Exponential Form): $\ln(e^{a}) = a$
  • Inverse Rule (both side to power of $c$): $e^{c \ln(a)} = a^{c}$

Exponential of $a^{x}$ In calculus, we often encounter the exponential of $a^{x}$ where $a$ is a constant. However, we usually write the exponential of $a^{x}$ as $e^{x \ln(a)}$.

This is signifies that the exponential of $a^{x}$ is equivalent to the exponential of $e$ to some constant times the $x$.

$$ a^{x} = e^{x \ln(a)} = e^{x \times \text{constant}}$$

This way of writing is not only limited to base of $e$, we can write the exponential of $a^{x}$ as $b^{x \log_{b}(a)}$ where $b$ is any base. For example:

$$a^{x} = b^{x \log_{b}(a)} = e^{x \ln(a)} = \pi^{x \log_{\pi}(a)} = 42^{x \log_{42}(a)} = \ldots$$