Logarithms
Formula Sheet 5
1. Definition & Basics
The Definition:
If $$ x = a^y $$ , then $$ \log_{a}x = y $$
Constraints (Must Remember):
- Base $$ a > 0 $$ and $$ a \neq 1 $$
- Value $$ x > 0 $$
- $$ y $$ can be any real number
Standard Bases:
• Common: $$ \log_{10}x $$
• Natural: $$ \log_{e}x $$ (ln x)
• Binary: $$ \log_{2}x $$
• Common: $$ \log_{10}x $$
• Natural: $$ \log_{e}x $$ (ln x)
• Binary: $$ \log_{2}x $$
2. Logarithm Properties
| Property | Formula |
|---|---|
| Product Rule | $$ \log_a(xy) = \log_a x + \log_a y $$ |
| Quotient Rule | $$ \log_a(\frac{x}{y}) = \log_a x – \log_a y $$ |
| Power Rule | $$ \log_a(x^m) = m \cdot \log_a x $$ |
| Identity Rules |
$$ \log_a a = 1 $$ $$ \log_a 1 = 0 $$ |
| Inverse Rule | $$ b^{\log_b x} = x $$ |
| Reciprocal Arg | $$ -\log_a x = \log_a (\frac{1}{x}) $$ |
3. Base Changing Formulas
Reciprocal Base
$$ \log_a b = \frac{1}{\log_b a} $$
New Base ‘c’
$$ \log_a b = \frac{\log_c b}{\log_c a} $$
⚠️ Common Silly Mistakes
The following are FALSE statements. Do not make these errors!
✕ $$ \log_a(\frac{x}{y}) \neq \log_a(x-y) $$
✕ $$ \log_a(xy) \neq (\log_a x)(\log_a y) $$
✕ $$ \log_a(x+y) \neq \log_a x + \log_a y $$
✕ $$ \log_a x^n \neq (\log_a x)^n $$
✕ $$ \frac{\log_u x}{\log_u y} \neq \log_u x – \log_u y $$
Notation Alert:
$$ \log^2_a x $$ means $$ (\log_a x)^2 $$
It is not $$ \log_a(x^2) $$
$$ \log^2_a x $$ means $$ (\log_a x)^2 $$
It is not $$ \log_a(x^2) $$