I am learning logarithms and i found that $log(a*b) = log(a)+log(b)$ I tried to apply the same principle for three numbers like $log(a*b*c) = log(a)+log(b)+log(c)$ but it didn't work as i expected.
Is there any direct formula to calculate this without multiplying the numbers? And also is it possible to extend this answer so that i can find $log(a_1*a_2*a_3*...*a_n)$ for any large value of $n$ for which i definitely can't multiply all the numbers to find the logarithm?
Given a finite collection $\{a_1,a_2,\cdots,a_N\}$ of positive numbers, it is true that $$\log \prod_{n=1}^N a_n=\sum_{n=1}^N \log a_n$$ I.e., $$\log(a_1a_2\cdots a_n)=\log a_1+\log a_2 +\cdots+\log a_N$$
In fact, $$\log(a_1^{p_1}a_2^{p_2} \cdots a_n^{p_n})= p_1\log a_1+p_2\log a_2 +\cdots+p_N\log a_N$$