Question

What role do logarithms play in the analysis of electronic circuits?

Answer 1

A logarithmic function is the inverse of an exponential function. If x^a=yxa=y, then logx(y)=alogx⁡(y)=a. If x=10x=10, it is an estimate of how many digits you need to write the number. (For example, log10(1000)=3log10⁡(1000)=3, and log101000000=6log10⁡1000000=6.

There are many uses for it. It has some nice properties. One really important one is that it turns products into sums. To see this, you can verify that logx(a⋅b)=log(a)+log(b)logx⁡(a⋅b)=log⁡(a)+log⁡(b). This means that, when it is hard to calculate a product, you can take the logarithms of all the factors and add them together, and then take the exponential function of the sum to get the product. This is how slide rules worked, which was what engineers used before pocket calculators.

In computer science, logarithms are used when you have to multiply a lot of small numbers, for example probabilities. Multiplying many small numbers together is not trivial since the results keep getting smaller, and the computer has bounds on its precision (if a number gets too small, it just gets rounded to 0). If you have two such products and you want to know which one is larger, it is much better to take the logarithm of both, since this way you will get more manageable numbers, which are safer to compare.

Logarithms are useful because they are the way our brain naturally understands most things. If there are two similar items that cost $2 and $4, then the $4 is a lot more expensive than the $2 item. You would probably buy the $2 item. If you have two items that cost $100 and $102, then the difference isn't much, and you might buy the more expensive one. The gap between these two is the same, but you have to multiply 2 once to get to 2, and twice to get to 4 - that's a whole extra two! .You have to multiply two 6.6 times to get to 100, and 6.7 times to get to 102 - that's a difference of only 0.1 twos when you think in logarithms.