Question

Can yo please explain the last part of this problem below

Answer 1

`\stackrel{f(x)}{500(0.97)^x}~~ = ~~\stackrel{g(x)}{20(1.03)^x}\implies \cfrac{500(0.97)^x}{20}=1.03^x\implies 25(0.97)^x=1.03^x \\\\\\ \cfrac{25(0.97)^x}{1.03^x}=1\implies \cfrac{0.97^x}{1.03^x}=\cfrac{1}{25}\implies \left( \cfrac{0.97}{1.03} \right)^x=\cfrac{1}{25}`

`\log\left[ \left( \cfrac{0.97}{1.03} \right)^x \right]=\log\left( \cfrac{1}{25} \right)\implies x\log\left( \cfrac{0.97}{1.03} \right)=\log\left( \cfrac{1}{25} \right) \\\\\\ x=\cfrac{\log\left( (1)/(25) \right)}{\log\left( (0.97)/(1.03) \right)}\implies \boxed{x\approx 54}`