1. The problem is to evaluate $\log_{10}\left(\frac{1}{10}\right)$ without using a calculator. 2. Recall the logarithm rule: $\log_b\left(\frac{1}{a}\right) = -\log_b(a)$ for any positive $a$ and base $b$. 3. Applying this rule, we have: $$\log_{10}\left(\frac{1}{10}\right) = -\log_{10}(10)$$ 4. Since $\log_{10}(10) = 1$ (because $10^1 = 10$), it follows that: $$-\log_{10}(10) = -1$$ 5. Therefore, the value of $\log_{10}\left(\frac{1}{10}\right)$ is: $$\boxed{-1}$$ This means the logarithm of the reciprocal of 10 to the base 10 is $-1$ because the exponent needed to get $\frac{1}{10}$ from 10 is $-1$.