প্রমাণ কর যে, $\frac{\log \sqrt{Q}+\log \sqrt{Y}-\log Z }{\log \left(2Y\right)-\log Z^2}=\frac{1}{2}$

দেওয়া আছে,

\(A = 2\), \(B = 3\), \(C = 5\) এবং \(D = 7\)

প্রমাণ করতে হবে যে,

\(D \log \frac{AC}{B^2} - A \log \frac{C^2}{A^2 B} + B \log \frac{B^4}{A^4 C} = -\log 2\)

বামপক্ষ:

\(= D \log \frac{AC}{B^2} - A \log \frac{C^2}{A^2 B} + B \log \frac{B^4}{A^4 C}\)

মান বসিয়ে পাই,

\(= 7 \log \frac{2 \times 5}{3^2} - 2 \log \frac{5^2}{2^2 \times 3} + 3 \log \frac{3^4}{2^4 \times 5}\)

\(= \log \left(\frac{2 \times 5}{3^2}\right)^7 - \log \left(\frac{5^2}{2^2 \times 3}\right)^2 + \log \left(\frac{3^4}{2^4 \times 5}\right)^3\)     [\(\because n \log x = \log x^n\)]

\(= \log \left(\frac{2^7 \times 5^7}{3^{14}}\right) - \log \left(\frac{5^4}{2^4 \times 3^2}\right) + \log \left(\frac{3^{12}}{2^{12} \times 5^3}\right)\)

\(= \log \left( \frac{\frac{2^7 \times 5^7}{3^{14}} \times \frac{3^{12}}{2^{12} \times 5^3}}{\frac{5^4}{2^4 \times 3^2}} \right)\)     [\(\because \log M - \log N + \log P = \log \left(\frac{M \cdot P}{N}\right)\)]

\(= \log \left( \frac{2^7 \times 5^7 \times 3^{12}}{3^{14} \times 2^{12} \times 5^3} \times \frac{2^4 \times 3^2}{5^4} \right)\)

\(= \log \left( \frac{2^{7+4} \times 3^{12+2} \times 5^7}{2^{12} \times 3^{14} \times 5^{3+4}} \right)\)

\(= \log \left( \frac{2^{11} \times 3^{14} \times 5^7}{2^{12} \times 3^{14} \times 5^7} \right)\)

\(= \log \left( 2^{11-12} \times 3^{14-14} \times 5^{7-7} \right)\)

\(= \log \left( 2^{-1} \times 3^0 \times 5^0 \right)\)

\(= \log \left( 2^{-1} \times 1 \times 1 \right)\)

\(= \log \left( 2^{-1} \right)\)

\(= -1 \cdot \log 2\)

\(= -\log 2\)

\(= \text{ডানপক্ষ}\)

অতএব, \(D \log \frac{AC}{B^2} - A \log \frac{C^2}{A^2 B} + B \log \frac{B^4}{A^4 C} = -\log 2\) (প্রমাণিত)