Solve 0.05=2*100*e^(-2*0.1^2N)

$$0.05 = 2 \times 100 \times e ^ { ( - 2 \times 0.1 ^ { 2 } N ) }$$

Solve for N

$$N=50\ln(4000)\approx 414.702482005$$

Steps Using Definition of Logarithm

Multiply 2 and 100 to get 200.

$$0.05=200e^{-2\times 0.1^{2}N}$$

Calculate 0.1 to the power of 2 and get 0.01.

$$0.05=200e^{-2\times 0.01N}$$

Multiply -2 and 0.01 to get -0.02.

$$0.05=200e^{-0.02N}$$

Swap sides so that all variable terms are on the left hand side.

$$200e^{-0.02N}=0.05$$

Divide both sides by 200.

$$e^{-0.02N}=\frac{0.05}{200}$$

Expand \frac{0.05}{200} by multiplying both numerator and the denominator by 100.

$$e^{-0.02N}=\frac{5}{20000}$$

Reduce the fraction \frac{5}{20000} to lowest terms by extracting and canceling out 5.

$$e^{-0.02N}=\frac{1}{4000}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-0.02N})=\log(\frac{1}{4000})$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$-0.02N\log(e)=\log(\frac{1}{4000})$$

Divide both sides by \log(e).

$$-0.02N=\frac{\log(\frac{1}{4000})}{\log(e)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$-0.02N=\log_{e}\left(\frac{1}{4000}\right)$$

Multiply both sides by -50.

$$N=-\frac{\ln(4000)}{-0.02}$$