Solve 3e^2-3n+3=54

$$3 e ^ { 2 - 3 n } + 3 = 54$$

Solve for n

$$n=\frac{2-\ln(17)}{3}\approx -0.277737781$$

Steps Using Definition of Logarithm

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

$$3e^{-3n+2}+3=54$$

Subtract 3 from both sides of the equation.

$$3e^{-3n+2}=51$$

Divide both sides by 3.

$$e^{-3n+2}=17$$

Take the logarithm of both sides of the equation.

$$\log(e^{-3n+2})=\log(17)$$

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

$$\left(-3n+2\right)\log(e)=\log(17)$$

Divide both sides by \log(e).

$$-3n+2=\frac{\log(17)}{\log(e)}$$

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

$$-3n+2=\log_{e}\left(17\right)$$

Subtract 2 from both sides of the equation.

$$-3n=\ln(17)-2$$

Divide both sides by -3.

$$n=\frac{\ln(17)-2}{-3}$$

Solve for n (complex solution)

$$n=-\frac{2\pi n_{1}i}{3}-\frac{\ln(17)}{3}+\frac{2}{3}$$

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