Solve 3e^-(2x+1)=10

$$3 e ^ { - ( 2 x + 1 ) } = 10$$

Solve for x

$$x=\frac{\ln(\frac{3}{10})-1}{2}\approx -1.101986402$$

Steps Using Definition of Logarithm

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

$$3e^{-\left(2x+1\right)}=10$$

Divide both sides by 3.

$$e^{-\left(2x+1\right)}=\frac{10}{3}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-\left(2x+1\right)})=\log(\frac{10}{3})$$

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

$$-\left(2x+1\right)\log(e)=\log(\frac{10}{3})$$

Divide both sides by \log(e).

$$-\left(2x+1\right)=\frac{\log(\frac{10}{3})}{\log(e)}$$

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

$$-\left(2x+1\right)=\log_{e}\left(\frac{10}{3}\right)$$

Divide both sides by -1.

$$2x+1=\frac{\ln(\frac{10}{3})}{-1}$$

Subtract 1 from both sides of the equation.

$$2x=\ln(\frac{3}{10})-1$$

Divide both sides by 2.

$$x=\frac{\ln(\frac{3}{10})-1}{2}$$

Solve for x (complex solution)

$$x=-\pi n_{1}i+\frac{\ln(\frac{3}{10})-1}{2}$$

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