Solve e^4x+1-3=26

$$e ^ { 4 x + 1 } - 3 = 26$$

Solve for x

$$x=\frac{\ln(29)-1}{4}\approx 0.591823957$$

Steps Using Definition of Logarithm

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

$$e^{4x+1}-3=26$$

Add 3 to both sides of the equation.

$$e^{4x+1}=29$$

Take the logarithm of both sides of the equation.

$$\log(e^{4x+1})=\log(29)$$

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

$$\left(4x+1\right)\log(e)=\log(29)$$

Divide both sides by \log(e).

$$4x+1=\frac{\log(29)}{\log(e)}$$

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

$$4x+1=\log_{e}\left(29\right)$$

Subtract 1 from both sides of the equation.

$$4x=\ln(29)-1$$

Divide both sides by 4.

$$x=\frac{\ln(29)-1}{4}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{2}+\frac{\ln(29)-1}{4}$$

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