Solve e^4x-2=28

$$e ^ { 4 x - 2 } = 28$$

Solve for x

$$x = \frac{\log_{e} {(28)} + 2}{4} \approx 1.333051128$$

Steps Using Definition of Logarithm

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

$$e^{4x-2}=28$$

Take the logarithm of both sides of the equation.

$$\log(e^{4x-2})=\log(28)$$

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

$$\left(4x-2\right)\log(e)=\log(28)$$

Divide both sides by \log(e).

$$4x-2=\frac{\log(28)}{\log(e)}$$

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

$$4x-2=\log_{e}\left(28\right)$$

Add 2 to both sides of the equation.

$$4x=\ln(28)-\left(-2\right)$$

Divide both sides by 4.

$$x=\frac{\ln(28)+2}{4}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{2}+\frac{\ln(28)+2}{4}$$

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