Question

evaluate the common logarithm without the use of a calculator.
log0.0001 =

Explanation:

Step1: Recall log definition

The common logarithm \(\log x\) is base - 10, so \(\log_{10}x = y\) means \(10^{y}=x\). We need to find \(y\) such that \(10^{y}=0.0001\).

Step2: Rewrite 0.0001 as power of 10

We know that \(0.0001=\frac{1}{10000}\), and \(10000 = 10^{4}\), so \(\frac{1}{10000}=10^{- 4}\) (since \(a^{-n}=\frac{1}{a^{n}}\) for \(a
eq0\) and \(n\in\mathbb{R}\)).

Step3: Solve for y

Since \(10^{y}=0.0001 = 10^{-4}\), by the property of exponential functions (if \(a^{m}=a^{n}\), then \(m = n\) for \(a>0,a
eq1\)), we have \(y=-4\).

Answer:

\(- 4\)