Question

What is the derivative of log10x with respect to logx?

Answer 1

Answer:

The answer is

y'=log10(e)⋅1x

Solution

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

loga(b)=loga(e)⋅loge(b)

Similarly, function log10(x) can be written as:

y=log10(e)⋅loge(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 log10(e) is constant, so only remaining function is loge(x)

Hence:

y'=log10(e)⋅1x

Alternate solution:

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

loga(b)=ln(b)ln(a)

From change of base we have log10(x)=log10(x)=ln(x)ln(10