Solve 2.3e^0.002t=4.6

$$2.3 { e }^{ 0.002t } =4.6$$

Solve for t

$$t=500\ln(2)\approx 346.57359028$$

Steps Using Definition of Logarithm

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

$$2.3e^{0.002t}=4.6$$

Divide both sides of the equation by 2.3, which is the same as multiplying both sides by the reciprocal of the fraction.

$$e^{0.002t}=2$$

Take the logarithm of both sides of the equation.

$$\log(e^{0.002t})=\log(2)$$

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

$$0.002t\log(e)=\log(2)$$

Divide both sides by \log(e).

$$0.002t=\frac{\log(2)}{\log(e)}$$

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

$$0.002t=\log_{e}\left(2\right)$$

Multiply both sides by 500.

$$t=\frac{\ln(2)}{0.002}$$