# Logarithmic Arithmetic Derivative

$\Large \frac{\frac{d}{dx}n}{n} \neq 0?$

In calculus, when you take the derivative of a constant you get zero as an answer. In number theory, there is something called the arithmetic derivative which allows you to differentiate a number and get a nonzero answer. The arithmetic derivative works as follows.

Where $$n'$$ denotes the arithmetic derivative of $$n$$:

$$p' = 1$$ for all primes $$p$$

$$(ab)'=a'b+ab'$$

$$0'=1'=0$$

For example, $$6'=(2\times3)'=(2')(3)+(2)(3')=(1)(3)+(2)(1)=5$$

Let us define the Logarithmic Arithmetic Derivative, denoted by $$L(n)$$, as $$\frac{n'}{n}$$. For how many ordered pairs $$(a,b)$$, where $$a$$ and $$b$$ are distinct positive integers less than $$100$$, does $$L(a)=L(b)$$?

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