Solve 3e^2x=10

$$3 e ^ { 2 x } = 10$$

Solve for x

$$x=\frac{\ln(\frac{10}{3})}{2}\approx 0.601986402$$

Steps Using Definition of Logarithm

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

$$3e^{2x}=10$$

Divide both sides by 3.

$$e^{2x}=\frac{10}{3}$$

Take the logarithm of both sides of the equation.

$$\log(e^{2x})=\log(\frac{10}{3})$$

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

$$2x\log(e)=\log(\frac{10}{3})$$

Divide both sides by \log(e).

$$2x=\frac{\log(\frac{10}{3})}{\log(e)}$$

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

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

Divide both sides by 2.

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

Solve for x (complex solution)

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

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