Understanding and Expanding the 2nd Chebyshev Function: Calculation Help

[camilus]

I'm a bit confused as to how to calculate the 2nd chebyshev function.

I know $$\psi (x) = \sum_{p^k \le x} \ln p$$

but can someone show me how to expand it? Like, do I use for 2^k, 2, 4, 8, 16, 32... for all the $$p^k \le x$$, same with 3, and 4 and so on?

MW gives the example:

$$\psi (10) = \ln (2520) = 3\ln2+2\ln3+\ln5+\ln7$$

how would say, $$\psi (30)$$ be written?

 

[g_edgar]

$$\spi(30)$$
Let's see...
$$2,4,8,16 < 30$$
$$3,9,27 < 30$$
$$5,25 < 30$$
$$7 < 30$$
$$11 < 30$$
$$13 < 30$$
$$17 < 30$$
$$19 < 30$$
$$23 < 30$$
$$29 < 30$$
so ... $$\psi(30) = 4\log 2 + 3\log 3 + 2\log 5 + \log 7 + \log 11 + \log 13 + \log 17 + \log 19 + \log 23 + \log 29$$