Solve 39.6=63.3*e^-ke(9.5)

$$39.6 = 63.3 \times e ^ { - k e } ( 9.5 )$$

Solve for k

$$k=\frac{\ln(\frac{4009}{264})}{e}\approx 1.000760106$$

Steps Using Definition of Logarithm

Multiply 63.3 and 9.5 to get 601.35.

$$39.6=601.35e^{\left(-k\right)e}$$

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

$$601.35e^{\left(-k\right)e}=39.6$$

Divide both sides by 601.35.

$$e^{\left(-k\right)e}=\frac{39.6}{601.35}$$

Expand \frac{39.6}{601.35} by multiplying both numerator and the denominator by 100.

$$e^{\left(-k\right)e}=\frac{3960}{60135}$$

Reduce the fraction \frac{3960}{60135} to lowest terms by extracting and canceling out 15.

$$e^{\left(-k\right)e}=\frac{264}{4009}$$

Reorder the terms.

$$e^{-ek}=\frac{264}{4009}$$

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

$$e^{\left(-e\right)k}=\frac{264}{4009}$$

Take the logarithm of both sides of the equation.

$$\log(e^{\left(-e\right)k})=\log(\frac{264}{4009})$$

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

$$\left(-e\right)k\log(e)=\log(\frac{264}{4009})$$

Divide both sides by \log(e).

$$\left(-e\right)k=\frac{\log(\frac{264}{4009})}{\log(e)}$$

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

$$\left(-e\right)k=\log_{e}\left(\frac{264}{4009}\right)$$

Divide both sides by -e.

$$k=\frac{\ln(\frac{264}{4009})}{-e}$$