In the steps to calculate maximum likelihood we often re-write a product expression as a sum expression.
I think I understand the following equation $$ \prod_{i=1}^n p^{y_i} = p^{\sum_{i=1}^n y_i } $$ which makes use of the rule $$ x^a x^b = x^{a+b}$$
However I am having trouble working how to re-write the following as a sum.
Where b is a constant.
How do I do that?
$$
\prod_{i=1}^n x_i^{1+b}
$$
[Update]
As advised I cannot combine the exponents due to different bases. So I need to take the log which is the next step in MLE anyway.
$Log ( x_1^{1+b} x_2^{1+b} ... x_n^{1+b})$
But still I am not seeing how to convert to the sum.
I thought of moving (1+b) to be in front of the log using the rule $$log(x^b) = b \ log(a)$$ but I am not confident it works that way with different bases.
The powerful tool that is the logarithm works well even if bases are not the same. So, note that first $$x_1^{1+b} x_2^{1+b} \ldots x_n^{1+b} = \left(x_1 x_2 \ldots x_n \right)^{1+b}$$ so that
$$ \log \left(x_1 \cdot x_2 \cdot \ldots x_n \right)^{1+b} = (1+b) \sum_{i = 1}^n \log x_i. $$
and now you have a sum of logarithms.