For a space $X$ and a spectrum $E$, we define $X\wedge E$ as the spectrum with $n$th space $X\wedge E_n$ and obvious connecting maps. I wonder if the following identification holds in general:
$$H^n(X\wedge E; \mathbb{R})\cong H^n(X; H^*(E;\mathbb{R})).$$
I believe the left-hand side equals $$[X\wedge E, H\mathbb{R}]_{-n}$$ while RHS equals $$[X, K(H^{*}(E;\mathbb{R}),n)]$$ where $K(H^{*}(E;\mathbb{R}),n)$ is the $n$th space of the Eilenberg-MacLane spectrum $HH^{*}(E;\mathbb{R})$. I believe those two expressions are closely related but I could not see how to prove it.
Thank you in advance for any corrections, answers, and comments!