Question

How do you differentiate `log_2 (x)`?

Answer 1

`d/dx log_2(x) = 1/(x*ln(2))`

Explanation:

This follows from the general formula:
`color(white)("XXX")d/dx(log_a(x)) = 1/(x*ln(a))`

Answer 2

`"d"/("d"x) [log_2(x)] = 1/(xln(2))`

Explanation:

As we know how to differentiate `ln(x)`, we should change the base of the logarithm first.

The according formula to change a logarithmic expression from the base `a` to the base `b` is

`log_color(red)(a)(color(blue)(x)) = log_b(color(blue)(x)) / log_b(color(red)(a))`

You can apply the formula as follows:

`log_2(x) = ln(x) / ln(2)`

As `1/ln(2)` is just a constant and the derivative of `ln(x)` is `1/x`, our derivative is:

`"d"/("d"x) [log_2(x)] = "d"/("d"x) [ln(x) / ln(2) ] = 1/ln(2) * 1/x = 1/(xln(2))`