Solve e^2x-5+1=4

$$e ^ { 2 x - 5 } + 1 = 4$$

Solve for x

$$x = \frac{\log_{e} {(3)} + 5}{2} \approx 3.049306144$$

Steps Using Definition of Logarithm

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

$$e^{2x-5}+1=4$$

Subtract 1 from both sides of the equation.

$$e^{2x-5}=3$$

Take the logarithm of both sides of the equation.

$$\log(e^{2x-5})=\log(3)$$

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

$$\left(2x-5\right)\log(e)=\log(3)$$

Divide both sides by \log(e).

$$2x-5=\frac{\log(3)}{\log(e)}$$

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

$$2x-5=\log_{e}\left(3\right)$$

Add 5 to both sides of the equation.

$$2x=\ln(3)-\left(-5\right)$$

Divide both sides by 2.

$$x=\frac{\ln(3)+5}{2}$$

Solve for x (complex solution)

$$x=\frac{\ln(3)+5}{2}+i\pi n_{1}$$

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