1. **State the problem:** Express $\log_2 20$ in terms of $\log_{10_k}$ given the relation $\log_a \tau = \frac{\log_a x}{\log_a a}$. 2. **Recall the change of base formula:** For any positive numbers $a,b,c$ with $a \neq 1$ and $b \neq 1$, $$\log_b x = \frac{\log_a x}{\log_a b}$$ This formula allows us to express logarithms in one base in terms of logarithms in another base. 3. **Apply the formula to $\log_2 20$:** Using base $10_k$ (which we interpret as base $10_k$), $$\log_2 20 = \frac{\log_{10_k} 20}{\log_{10_k} 2}$$ 4. **Explanation:** This means to find $\log_2 20$, you can take the logarithm of 20 in base $10_k$ and divide it by the logarithm of 2 in base $10_k$. 5. **Final answer:** $$\boxed{\log_2 20 = \frac{\log_{10_k} 20}{\log_{10_k} 2}}$$ This expresses $\log_2 20$ in terms of $\log_{10_k}$ as requested.