Solve e^frac{x{4}}=205

$$e ^ { \frac { x } { 4 } } = 205$$

Solve for x

$$x=4\ln(205)\approx 21.292039917$$

Steps Using Definition of Logarithm

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

$$e^{\frac{1}{4}x}=205$$

Take the logarithm of both sides of the equation.

$$\log(e^{\frac{1}{4}x})=\log(205)$$

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

$$\frac{1}{4}x\log(e)=\log(205)$$

Divide both sides by \log(e).

$$\frac{1}{4}x=\frac{\log(205)}{\log(e)}$$

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

$$\frac{1}{4}x=\log_{e}\left(205\right)$$

Multiply both sides by 4.

$$x=\frac{\ln(205)}{\frac{1}{4}}$$

Solve for x (complex solution)

$$x=4\ln(205)+i\times 8\pi n_{1}$$

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