Solve 18+6e^3-x/6+4=14

$$\frac { 18 + 6 e ^ { 3 - x } } { 6 } + 4 = 14$$

Solve for x

$$x=3-\ln(7)\approx 1.054089851$$

Steps Using Definition of Logarithm

Multiply both sides of the equation by 6.

$$18+6e^{3-x}+24=84$$

Add 18 and 24 to get 42.

$$42+6e^{3-x}=84$$

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

$$6e^{-x+3}+42=84$$

Subtract 42 from both sides of the equation.

$$6e^{-x+3}=42$$

Divide both sides by 6.

$$e^{-x+3}=7$$

Take the logarithm of both sides of the equation.

$$\log(e^{-x+3})=\log(7)$$

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

$$\left(-x+3\right)\log(e)=\log(7)$$

Divide both sides by \log(e).

$$-x+3=\frac{\log(7)}{\log(e)}$$

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

$$-x+3=\log_{e}\left(7\right)$$

Subtract 3 from both sides of the equation.

$$-x=\ln(7)-3$$

Divide both sides by -1.

$$x=\frac{\ln(7)-3}{-1}$$

Solve for x (complex solution)

$$x=-2\pi n_{1}i+3-\ln(7)$$

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