For positive integers $x$ and $y$ where $x\lt 8$, $\log_x8=y$. What is the value of $y$ ?

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Rewrite the logarithmic equation in exponential form.

$$\log_x8=y$$ $$x^y = 8$$

We can solve for $x$.

$$x = \sqrt[y]{8}$$

Then substitute:

$$x \lt 8$$ $$\sqrt[y]{8} \lt 8$$

Since $y$ and $x$ are integers, the only reasonable options are:

$$\sqrt[3]{8} = 2$$ $$\sqrt[1]{8} = 8$$

Since the number needs to be smaller than 8, only $\sqrt[3]{8}$ works, so $y=3$.