Solve e^-3r=21/203

$${ e }^{ -3r } = \frac{ 21 }{ 203 }$$

Solve for r

$$r=\frac{\ln(\frac{29}{3})}{3}\approx 0.756227847$$

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

$$e^{-3r}=\frac{3}{29}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-3r})=\log(\frac{3}{29})$$

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

$$-3r\log(e)=\log(\frac{3}{29})$$

Divide both sides by \log(e).

$$-3r=\frac{\log(\frac{3}{29})}{\log(e)}$$

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

$$-3r=\log_{e}\left(\frac{3}{29}\right)$$

Divide both sides by -3.

$$r=\frac{\ln(\frac{3}{29})}{-3}$$