Question

What is the derivative of `y=log_10(x)`?

Answer 1

The answer is

`y'=log_10(e)*1/x`

Solution

Suppose we have `log_a(b)`, we want to change it on exponential (e) base, then it can be written as:

`log_a(b)=log_a(e)*log_e(b)`

Similarly, function `log_10(x)` can be written as:

`y=log_10(e)*log_e(x)`

Let's say we have, `y=c*f(x)`, where c is a constant
then, `y'=c*f'(x)`

Now, this is quite straightforward to differentiate, as `log_10(e)` is constant, so only remaining function is `log_e(x)`

Hence:

`y'=log_10(e)*1/x`

Alternate solution:

Another common approach is to use the change of base formula, which says that:

`log_a(b) =ln(b)/ln(a)`

From change of base we have `log_10(x) = log_10(x) = ln(x)/ln(10)`.

This we can differentiate as long as we remember that

`1/ln(10)` is just a constant multipler.

Doing the problem this way gives a result of `y' = 1/ln(10)*1/x`.