The value of $\log_{10} (.0001)$ is

Concept & Formula To evaluate logarithms involving decimals, convert the decimal into a fraction and then into a negative power of the base. $$ \log_a (a^n) = n $$ Step-by-Step Solution * **Given:** Evaluate $\log_{10} (.0001)$. * **Calculation:** First, express the decimal $.0001$ as a fraction. Since there are four decimal places, it becomes: $$ \frac{1}{10000} $$ * Convert the denominator to a power of $10$: $$ \frac{1}{10^4} $$ * Bring the power to the numerator by making the exponent negative: $$ 10^{-4} $$ * Substitute this back into the logarithmic expression: $$ \log_{10} (10^{-4}) $$ * Using the power rule, the base and log cancel out, leaving the exponent: $$ -4 $$ Exam Strategy & Shortcut For any base $10$ logarithm of a decimal like $0.00...1$, simply count the number of decimal places. $0.0001$ has $4$ decimal places. Therefore, it is $10^{-4}$, and the logarithm is instantly $-4$. Common Pitfall A common error is missing the negative sign and answering $4$, confusing $.0001$ with $10000$. Remember that fractional values (between $0$ and $1$) yield negative logarithms when the base is greater than $1$. Final Answer **Therefore, the correct answer is -4.**