Solve e^-5+3t=1

$$e ^ { - 5 + 3 t } = 1$$

Solve for t

$$t = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

Steps Using Definition of Logarithm

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

$$e^{3t-5}=1$$

Take the logarithm of both sides of the equation.

$$\log(e^{3t-5})=\log(1)$$

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

$$\left(3t-5\right)\log(e)=\log(1)$$

Divide both sides by \log(e).

$$3t-5=\frac{\log(1)}{\log(e)}$$

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

$$3t-5=\log_{e}\left(1\right)$$

Add 5 to both sides of the equation.

$$3t=-\left(-5\right)$$

Divide both sides by 3.

$$t=\frac{5}{3}$$