To rewrite \(\log\left(\frac{1}{100}\right)\) using the properties of logarithms, specifically utilizing the Power Rule of Logarithms, we can proceed as follows:
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Start by expressing \(\frac{1}{100}\) in another form: \[ \frac{1}{100} = 100^{-1} \]
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Now, apply the logarithm: \[ \log\left(\frac{1}{100}\right) = \log\left(100^{-1}\right) \]
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According to the Power Rule of Logarithms, which states that \(\log(a^k) = k \log(a)\), we can rewrite the logarithm: \[ \log\left(100^{-1}\right) = -1 \cdot \log(100) \] or \[ \log\left(\frac{1}{100}\right) = -\log(100) \]
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Next, we recognize that \(100\) can be expressed as \(10^2\): \[ \log(100) = \log(10^2) \]
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Again using the Power Rule: \[ \log(10^2) = 2 \log(10) \]
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Remember that \(\log(10) = 1\) (for base 10 logarithm): \[ \log(100) = 2 \cdot 1 = 2 \]
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Substitute this back into our expression: \[ \log\left(\frac{1}{100}\right) = -\log(100) = -2 \]
Thus, the rewritten form of \(\log\left(\frac{1}{100}\right)\) is: \[ \log\left(\frac{1}{100}\right) = -2 \]