Introduction;
Logarithm and Exponent

Computing with large and small numbers

Computations work well when numbers are

• of similar magnitude:
  10, 20, 15, 32, 1, 5
  but not
  1, 100000000, 0.00000000001
• Not too large and not too small:
  $10^{-10}$ and $10^{10}$
  but not
  $10^{-1000}$ and $10^{1000}$

Computing with large and small numbers

• What if we need to compute $10000! = \prod_{i=1}^{10000} i$?
• How about analyzing a sequence $a_i = \exp(i^2)$ for $i \in \{1, ..., 1000\}$?
• We can compute on logarithmic scale! and then exponentiate the final result if needed

Logarithm: definition

Logarithm is a function $\mathbb{R}^+ \rightarrow \mathbb{R}$: $$\log_b x = y$$

• b — base (בסיס), $b > 1$
• y - power (חזקה)

By definition, $$b^y = x$$

Logarithm: shape

• defined on $\mathbb{R}^+$
• maps to $\mathbb{R}$
• increasing and concave

Logarithm: properties

• $ \log 1 = 0$
• $ \log x\cdot y = \log x + \log y$
• $ \log \frac 1 x = - \log x$
• $ \log x^a = a \log x$

Logarithm: bases

$\log x$ assumes a base (binary, natural, decimal):

The base can be explicit: $\log_2 x$, $\log_e x$, $\log_{10} x$

Logarithm: bases