Alg9.4.36

If $(3.7)^x=(0.037)^y=10000\,$ then find the value of 1/x-1/y

$(3.7)^x=10000=10^4\,$

$x\log_{10} (3.7)=4\,$

Hence

$\frac{1}{x}=\frac{\log_{10}(3.7)}{4}\,$

The next one,

$(0.037)^y=10000=10^4\,$

$y\log_{10}(0.037)=4\,$

$\frac{1}{y}=\frac{\log_{10}(0.037)}{4}\,$

Hence

$\frac{1}{x}-\frac{1}{y}=\frac{1}{4}\cdot(\log_{10}(3.7)-\log_{10}(0.037))\,$

$\frac{1}{x}-\frac{1}{y}=\frac{1}{4}\cdot(\log_{10}(\frac{3.7}{0.037})\,$

$\frac{1}{x}-\frac{1}{y}=\frac{1}{4}\log_{10}(100)\,$

$\frac{1}{x}-\frac{1}{y}=\frac{1}{2}\,$