Solve e^3x-9=8

$$e ^ { 3 x - 9 } = 8$$

Solve for x

$$x=\ln(2)+3\approx 3.693147181$$

Steps Using Definition of Logarithm

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

$$e^{3x-9}=8$$

Take the logarithm of both sides of the equation.

$$\log(e^{3x-9})=\log(8)$$

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

$$\left(3x-9\right)\log(e)=\log(8)$$

Divide both sides by \log(e).

$$3x-9=\frac{\log(8)}{\log(e)}$$

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

$$3x-9=\log_{e}\left(8\right)$$

Add 9 to both sides of the equation.

$$3x=3\ln(2)-\left(-9\right)$$

Divide both sides by 3.

$$x=\frac{3\ln(2)+9}{3}$$

Solve for x (complex solution)

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

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