Solve 500={(e)^x}+x

$$500= {(e)^{ x }} +x$$

Solve for x

$$x=\ln(500)\approx 6.214608098$$

Steps Using Definition of Logarithm

Swap sides so that all variable terms are on the left hand side.

$$e^{x}=500$$

Take the logarithm of both sides of the equation.

$$\log(e^{x})=\log(500)$$

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

$$x\log(e)=\log(500)$$

Divide both sides by \log(e).

$$x=\frac{\log(500)}{\log(e)}$$

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

$$x=\log_{e}\left(500\right)$$

Solve for x (complex solution)

$$x=\ln(500)+2\pi n_{1}i$$

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