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 \(y=3\).