Using numerical analysis and the Power Rule of Logarithms, log x^k=k log x, how can log(1/100) be rewritten?

1 answer

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:

  1. Start by expressing \(\frac{1}{100}\) in another form: \[ \frac{1}{100} = 100^{-1} \]

  2. Now, apply the logarithm: \[ \log\left(\frac{1}{100}\right) = \log\left(100^{-1}\right) \]

  3. 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 \[ \log\left(\frac{1}{100}\right) = -\log(100) \]

  4. Next, we recognize that \(100\) can be expressed as \(10^2\): \[ \log(100) = \log(10^2) \]

  5. Again using the Power Rule: \[ \log(10^2) = 2 \log(10) \]

  6. Remember that \(\log(10) = 1\) (for base 10 logarithm): \[ \log(100) = 2 \cdot 1 = 2 \]

  7. 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: \[ \log\left(\frac{1}{100}\right) = -2 \]