Solve 4e^2x-4=6

$$4 e ^ { 2 x - 4 } = 6$$

Solve for x

$$x = \frac{\log_{e} {(\frac{3}{2})} + 4}{2} \approx 2.202732554$$

Steps Using Definition of Logarithm

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

$$4e^{2x-4}=6$$

Divide both sides by 4.

$$e^{2x-4}=\frac{3}{2}$$

Take the logarithm of both sides of the equation.

$$\log(e^{2x-4})=\log(\frac{3}{2})$$

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

$$\left(2x-4\right)\log(e)=\log(\frac{3}{2})$$

Divide both sides by \log(e).

$$2x-4=\frac{\log(\frac{3}{2})}{\log(e)}$$

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

$$2x-4=\log_{e}\left(\frac{3}{2}\right)$$

Add 4 to both sides of the equation.

$$2x=\ln(\frac{3}{2})-\left(-4\right)$$

Divide both sides by 2.

$$x=\frac{\ln(\frac{3}{2})+4}{2}$$

Solve for x (complex solution)

$$x=\frac{\ln(\frac{3}{2})+4}{2}+\pi n_{1}i$$

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