Question

How do you express `log_2 5` in terms of common logs?

Answer 1

`color(blue)((log_(10)5)/(log_(10)2)`

Explanation:

I am assuming by common logs this means base 10.

If:

`y=log_(b)a<=>b^y=a`

Suppose we wish to express this using a different base. Let's say to a base `bbc`.

From:

`b^y=a`

Take logarithms to the base `c` of both sides:

`ylog_(c)b=log_(c)a`

Divide by `log_(c)b`:

`y=(log_(c)a)/(log_(c)b)`

From above:

`y=log_(b)a`

`:.`

`log_(b)a=(log_(c)a)/(log_(c)b)`

This is the change of base formula:

`log_(2)5=(log_(10)5)/(log_(10)2`