Solve 15,000=15,000e^-kt

$$15,000 = 15,000 e ^ { - k t }$$

Solve for k

$$\left\{\begin{matrix}\\k=0\text{, }&\text{unconditionally}\\k\in \mathrm{R}\text{, }&t=0\end{matrix}\right.$$

Steps Using Definition of Logarithm

Divide both sides by 15000.

$$\frac{15000}{15000}=e^{\left(-k\right)t}$$

Divide 15000 by 15000 to get 1.

$$1=e^{\left(-k\right)t}$$

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

$$e^{\left(-k\right)t}=1$$

Reorder the terms.

$$e^{-kt}=1$$

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

$$e^{\left(-t\right)k}=1$$

Take the logarithm of both sides of the equation.

$$\log(e^{\left(-t\right)k})=\log(1)$$

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

$$\left(-t\right)k\log(e)=\log(1)$$

Divide both sides by \log(e).

$$\left(-t\right)k=\frac{\log(1)}{\log(e)}$$

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

$$\left(-t\right)k=\log_{e}\left(1\right)$$

Divide both sides by -t.

$$k=\frac{0}{-t}$$

Solve for t

$$\left\{\begin{matrix}\\t=0\text{, }&\text{unconditionally}\\t\in \mathrm{R}\text{, }&k=0\end{matrix}\right.$$

Steps Using Definition of Logarithm

Divide both sides by 15000.

$$\frac{15000}{15000}=e^{\left(-k\right)t}$$

Divide 15000 by 15000 to get 1.

$$1=e^{\left(-k\right)t}$$

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

$$e^{\left(-k\right)t}=1$$

Take the logarithm of both sides of the equation.

$$\log(e^{\left(-k\right)t})=\log(1)$$

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

$$\left(-k\right)t\log(e)=\log(1)$$

Divide both sides by \log(e).

$$\left(-k\right)t=\frac{\log(1)}{\log(e)}$$

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

$$\left(-k\right)t=\log_{e}\left(1\right)$$

Divide both sides by -k.

$$t=\frac{0}{-k}$$