A-level Physics (Advancing Physics)/Digital Storage/Logarithms

That formula involved logarithms. If ab = c, then:

$$\log _ac = b$$

In other words, a logarithm is a way of asking the question, "a to the power of what is c?" a is known as the base of the logarithm. log a is shorthand for log10a, and can be calculated using the 'log' button on a scientific calculator.

Just as with indices, there are some laws of logarithms. At present, we are only concerned with this one:

$$\log _a(c^n) = n \log _ac$$

If we apply this to the formula for calculating the number of possible values a pixel can take on (v = 2b), we can rearrange it to get a formula for b by taking the logarithm to the base 10 of both sides:

$$\log v = \log (2^b)$$

$$\log v = b \log 2$$

$$b = \frac{\log{v}}{\log{2}}$$

We can then use a scientific calculator to calculate log 2, giving us the formula:

$$b \approx \frac{\log{v}}{0.3}$$