Question

on what day will there be less than 100 left? following the equation y = 5000(0.5)x

Answer 1

The given expression is

$y=5000(0.5)^x$

To find the day, we have to evaluate the function where y < 100.

$\begin{gathered} 5000(0.5)^x<100 \\ (0.5)^x<\frac{100}{5000} \\ (0.5)^x<\frac{1}{50} \end{gathered}$

Then, apply a logarithm on each side

$\begin{gathered} \ln (0.5)^x<\ln (\frac{1}{50}) \\ x\cdot\ln (0.5)<\ln (\frac{1}{50}) \\ x<\frac{\ln(\frac{1}{50})}{\ln(0.5)} \\ x<5.6 \end{gathered}$

Therefore, on day 5 it will be less than 100.