Solve 2/3=e^t-2x

$$2 / 3 = e ^ { t - 2 x }$$

Solve for t

$$t=2x+\ln(\frac{2}{3})$$

Steps Using Definition of Logarithm

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

$$e^{t-2x}=\frac{2}{3}$$

Take the logarithm of both sides of the equation.

$$\log(e^{t-2x})=\log(\frac{2}{3})$$

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

$$\left(t-2x\right)\log(e)=\log(\frac{2}{3})$$

Divide both sides by \log(e).

$$t-2x=\frac{\log(\frac{2}{3})}{\log(e)}$$

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

$$t-2x=\log_{e}\left(\frac{2}{3}\right)$$

Subtract -2x from both sides of the equation.

$$t=\ln(\frac{2}{3})-\left(-2x\right)$$

Solve for x

$$x=\frac{t+\ln(\frac{3}{2})}{2}$$

Steps Using Definition of Logarithm

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

$$e^{t-2x}=\frac{2}{3}$$

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

$$e^{-2x+t}=\frac{2}{3}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-2x+t})=\log(\frac{2}{3})$$

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

$$\left(-2x+t\right)\log(e)=\log(\frac{2}{3})$$

Divide both sides by \log(e).

$$-2x+t=\frac{\log(\frac{2}{3})}{\log(e)}$$

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

$$-2x+t=\log_{e}\left(\frac{2}{3}\right)$$

Subtract t from both sides of the equation.

$$-2x=\ln(\frac{2}{3})-t$$

Divide both sides by -2.

$$x=\frac{-t+\ln(\frac{2}{3})}{-2}$$

Solve for t (complex solution)

$$t=2\pi n_{1}i+2x+\ln(\frac{2}{3})$$

$$n_{1}\in \mathrm{Z}$$

Solve for x (complex solution)

$$x=-\pi n_{1}i+\frac{t+\ln(\frac{3}{2})}{2}$$

$$n_{1}\in \mathrm{Z}$$