Question

What is `log(5*8)` equal to?

Answer 1

`log(5*8) = 2log(2)+1 ~~ 1.60206`

Explanation:

If `x, y > 0` then:

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

(This follows from `10^(a+b) = 10^a * 10^b` and the definition of common logarithms)

`color(white)()`
In general for any valid base `b` (i.e. `b > 0` and `b != 1`) we also have:

`log_b b = 1`

(This follows from `b^1 = b` and the definition of logarithms in general)

So in particular, for common (i.e. base `10`) logarithms we have:

`log 10 = 1`

`color(white)()`
It is also useful to know that:

`log 2 ~~ 0.30103`

This is a very good approximation. The actual value is nearer `0.301029995664`

`color(white)()`
Putting this all together, we find:

`log(5*8) = log(2*2*10)`

`color(white)(log(5*8)) = log(2)+log(2)+log(10)`

`color(white)(log(5*8)) = 2log(2)+1`

`color(white)(log(5*8)) ~~ 2*0.30103+1 = 1.60206`