Solve 4e^3x=9

$$4 e ^ { 3 x } = 9$$

Solve for x

$$x=\frac{2\ln(\frac{3}{2})}{3}\approx 0.270310072$$

Steps Using Definition of Logarithm

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

$$4e^{3x}=9$$

Divide both sides by 4.

$$e^{3x}=\frac{9}{4}$$

Take the logarithm of both sides of the equation.

$$\log(e^{3x})=\log(\frac{9}{4})$$

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

$$3x\log(e)=\log(\frac{9}{4})$$

Divide both sides by \log(e).

$$3x=\frac{\log(\frac{9}{4})}{\log(e)}$$

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

$$3x=\log_{e}\left(\frac{9}{4}\right)$$

Divide both sides by 3.

$$x=\frac{\ln(\frac{9}{4})}{3}$$

Solve for x (complex solution)

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

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