Is the derivative of ln(x) the same as log(x)?

Answer 1

The answer is no, the derivative for the natural log function is not the same as for common log.

If you can remember the change of base formula for logarithms, that is probably the easiest way to remember how to find the derivative for functions like f(x) = log_b(x).

Use change of base to convert log_b(x) in terms of natural log,

log_bx = ln(x)/ln(b) (this works for converting to any other base; for example to convert to base a: log_ax/log_ab).

For the posted problem:

f(x) = log₁₀x = ln(x)/ln(10) = [1/ln(10)] ln(x)

ln(10) is just a constant, so...

f'(x) = [1/ln(10)] (1/x) = 1/(xln(10)), x>0

In general if u is a function of x:

y = log_au

y' = (1/u)(du/dx)log_ae or (1/[uln(a)] (du/dx)

Easier to remember if you think "change of base: ln(top)/ln(bottom)".

y = ln(u)/ln(a) = [1/ln(a)] ln(u)

y' = [1/ln(a)] (1/u) (du/dx)

This graph shows the comparison between the two posted functions.

desmos.com/calculator/jb7xmyshdb