Example Question - equivalence

Logarithm Equivalence Proof

<p>Given \(\log\frac{81}{8} = 2\log\frac{2}{3} + 3\log\frac{1}{3}\)</p> <p>Apply logarithm properties: \(\log a^b = b \cdot \log a\)</p> <p>\(\log\frac{3^4}{2^3} = \log\left(\frac{2}{3}\right)^2 + \log\left(\frac{1}{3}\right)^3\)</p> <p>\(\log\frac{3^4}{2^3} = \log\left(\frac{2^2}{3^2}\right) + \log\left(\frac{1}{3^3}\right)\)</p> <p>\(\log\frac{3^4}{2^3} = \log\frac{2^2}{3^2} + \log\frac{1}{3^3}\)</p> <p>\(\log\frac{3^4}{2^3} = \log\frac{2^2}{3^2} + \log\frac{1}{3^3}\)</p> <p>Combine logs on the right using \(\log a + \log b = \log(ab)\):</p> <p>\(\log\frac{3^4}{2^3} = \log\left(\frac{2^2}{3^2} \cdot \frac{1}{3^3}\right)\)</p> <p>\(\log\frac{3^4}{2^3} = \log\frac{2^2}{3^5}\)</p> <p>Simplify the right side:</p> <p>\(\log\frac{3^4}{2^3} = \log\frac{3^4}{2^3}\)</p> <p>Since the logs with the same base are equal, their arguments must be equal, proving the statement:</p> <p>\(\frac{3^4}{2^3} = \frac{3^4}{2^3}\)</p> <p>Hence, the given equation is proved to be correct.</p>