Question

How do logarithms work?

Answer 1

See explanation...

Explanation:

Logarithms work because exponents work...

In general `10^a * 10^b = 10^(a+b)`

This is easier to see in the case where `a` and `b` are positive whole numbers:

`10^a xx 10^b = stackrel "a times" overbrace(10xx10xx...xx10) xx stackrel "b times" overbrace(10xx10xx...xx10)`

`=stackrel "a+b times" overbrace(10xx10xx...xx10) = 10^(a+b)`

but it works for any Real numbers `a` and `b`.

If `x = 10^a` then we call `a` the common logarithm of `x`.

To say that `x = 10^a` and `y = 10^b` is the same as saying
`a = log_10 x` and `b = log_10 y` and we find:

`log_10 x + log_10 y = log_10 (xy)`

since:

`10^(a + b) = 10^a * 10^b`