Solving for exponent
$$ \log_{b}(b^{X}) = X $$Change of base formula: do this when you are changing the number system, for example, you need to change from counting in binary to hexadecimal or from decimal to base e.
$$\log_{b}(A) = \frac{\log_{n}(A)}{\log_{n}(b)}$$Where n is the new target base
Multiplication property
$$ \log_{b}(X \times Y) = \log_{b}(X) + \log_{b}(Y) $$Division property
$$ \log_{b} {\left( \frac{X}{Y} \right) } = \log_{b}(X) - \log_{b}(Y) $$Exponent property
$$ \log_{b}{ \left( x^{n} \right) } = n\log_{b}(x) $$Useful series
$$ \ln(1 + x) = \sum^{\infty}_{n=0} \left( \frac{(-1)^{(n+1)}}{n} \right) $$NOTE on the series: if $x \lt 1$ then the series converges, else if $x \gt 1$ then the series diverges
Example: you need to translate each letter of the English alphabet into a list of cells whose cells may only contain the number 1 or 0, and cannot remain empty. How long should each list be if there needs to be at least 64 different combinations?
We can create a placeholder of the length of the list and call it L. The equation should be as follows:
$$ 2^{L} = 64 $$We then need to solve for L. To do this, we take $\log_{2}(2^{L}) = L = \log_{2}(64)$
You are probably not very good at counting in binary but you are probably good at counting in base 10, so changing the base becomes useful to solve this problem. Take the right hand side of the equation:
$$ \log_{2}(64) = \frac{\log(64)}{\log(2)} $$Now the equation should look like this:
$$ L = \frac{\log(64)}{\log(2)} $$Which is solvable:
$$ L = \frac{1.8}{0.3} $$ $$ L = 6 $$First column is the value in base 10, the second is the log base 10 of that value rounded. Notice that as it approaches 10, the log value rises similarly up to 1, then the pattern repeats to the next 10 digit place. This pattern is similar for the other bases. Similar effects with decimals. Remember, this is a way to prove the Kraft inequality of prefix-free codes.
| Number n in base 10 | $\log(n)$ |
|---|---|
| 1 | 0 |
| 2 | 0.3 |
| 3 | 0.5 |
| 4 | 0.6 |
| 5 | 0.7 |
| 6 | 0.8 |
| 7 | ~0.8 |
| 8 | ~0.9 |
| 9 | ~1 |
| 10 | 1 |
| 11 | ~1 |
| 12 | ~1.1 |
| 90 | ~1.95 |
| 100 | 2 |
| 0.5 | -0.3 |
| 0.1 | -1 |
| 0.01 | -2 |
| 0.001 | -3 |
This page gives some quick references for working in decibels.