Simplify \(1\times {10}^{-3}\) to \({10}^{-3}\).
\[\log{\frac{0.1}{{({10}^{-3})}^{2}}}\]
Use Negative Power Rule: \({x}^{-a}=\frac{1}{{x}^{a}}\).
\[\log{\frac{0.1}{{(\frac{1}{{10}^{3}})}^{2}}}\]
Simplify \({10}^{3}\) to \(1000\).
\[\log{\frac{0.1}{{(\frac{1}{1000})}^{2}}}\]
Use Division Distributive Property: \({(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}\).
\[\log{\frac{0.1}{\frac{1}{{1000}^{2}}}}\]
Simplify \({1000}^{2}\) to \(1000000\).
\[\log{\frac{0.1}{\frac{1}{1000000}}}\]
Simplify \(\frac{0.1}{\frac{1}{1000000}}\) to \(100000\).
\[\log{100000}\]
The base of 10 is implied.
\[\log_{10}{100000}\]
Use Definition of Common Logarithm: \({b}^{a}=x\) if and only if \(log_b(x)=a\)1. Ask: If \({10}^{x}=100000\), what is x?2. Answer: \(5\).
\[5\]
5
