Simplify - log$_{10}$ 0.00001.

Analysis

The question asks us to simplify the expression $-\log_{10} 0.00001$. To solve this, we need to express the decimal number $0.00001$ as a power of $10$ and then apply the laws of logarithms.

Concept

The logarithm of a number $x$ to the base $a$ is the exponent to which $a$ must be raised to get $x$. That is:

Additionally, remember that:

Step-by-Step Solution

1. Rewrite the decimal in index form: $0.00001$ has five decimal places, so it can be written as $10^{-5}$.

2. Substitute this into the expression: $-\log_{10} (0.00001) = -\log_{10} (10^{-5})$

3. Apply the logarithm power rule: The power rule states that $\log_b(x^n) = n \cdot \log_b(x)$. Therefore, $\log_{10} (10^{-5}) = -5 \cdot \log_{10} (10)$.

4. Simplify the base: Since $\log_{10} (10) = 1$, the expression becomes: $-(-5 \cdot 1) = -(-5)$

5. Final calculation: $-(-5) = 5$

Conclusion

The simplified value of the expression is $5$. Comparing this to the provided options, it matches option C.

Exam Tip: When dealing with decimals in logarithms, always convert them to standard form (scientific notation) first. For example, $0.001 = 10^{-3}$, $0.01 = 10^{-2}$, and $0.1 = 10^{-1}$. This makes the calculation much faster and reduces the risk of errors with decimal points!