Solve 4e^3x-15=360

$$4 e ^ { 3 x - 15 } = 360$$

Solve for x

$$x = \frac{\log_{e} {(90)} + 15}{3} \approx 6.499936557$$

Steps Using Definition of Logarithm

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

$$4e^{3x-15}=360$$

Divide both sides by 4.

$$e^{3x-15}=90$$

Take the logarithm of both sides of the equation.

$$\log(e^{3x-15})=\log(90)$$

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

$$\left(3x-15\right)\log(e)=\log(90)$$

Divide both sides by \log(e).

$$3x-15=\frac{\log(90)}{\log(e)}$$

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

$$3x-15=\log_{e}\left(90\right)$$

Add 15 to both sides of the equation.

$$3x=\ln(90)-\left(-15\right)$$

Divide both sides by 3.

$$x=\frac{\ln(90)+15}{3}$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{3}+\frac{\ln(90)+15}{3}$$

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