Solve 3e^2x+7=6

$$3 e ^ { 2 x + 7 } = 6$$

Solve for x

$$x=\frac{\ln(2)-7}{2}\approx -3.15342641$$

Steps Using Definition of Logarithm

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

$$3e^{2x+7}=6$$

Divide both sides by 3.

$$e^{2x+7}=2$$

Take the logarithm of both sides of the equation.

$$\log(e^{2x+7})=\log(2)$$

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

$$\left(2x+7\right)\log(e)=\log(2)$$

Divide both sides by \log(e).

$$2x+7=\frac{\log(2)}{\log(e)}$$

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

$$2x+7=\log_{e}\left(2\right)$$

Subtract 7 from both sides of the equation.

$$2x=\ln(2)-7$$

Divide both sides by 2.

$$x=\frac{\ln(2)-7}{2}$$

Solve for x (complex solution)

$$x=\frac{\ln(2)-7}{2}+\pi n_{1}i$$

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