Solve 2*e^3x+4=5

$$2 \cdot e ^ { 3 x + 4 } = 5$$

Solve for x

$$x=\frac{\ln(\frac{5}{2})-4}{3}\approx -1.027903089$$

Steps Using Definition of Logarithm

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

$$2e^{3x+4}=5$$

Divide both sides by 2.

$$e^{3x+4}=\frac{5}{2}$$

Take the logarithm of both sides of the equation.

$$\log(e^{3x+4})=\log(\frac{5}{2})$$

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

$$\left(3x+4\right)\log(e)=\log(\frac{5}{2})$$

Divide both sides by \log(e).

$$3x+4=\frac{\log(\frac{5}{2})}{\log(e)}$$

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

$$3x+4=\log_{e}\left(\frac{5}{2}\right)$$

Subtract 4 from both sides of the equation.

$$3x=\ln(\frac{5}{2})-4$$

Divide both sides by 3.

$$x=\frac{\ln(\frac{5}{2})-4}{3}$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{3}+\frac{\ln(\frac{5}{2})-4}{3}$$

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