Solve 5+6e^4x-8=12

$$5 + 6 e ^ { 4 x - 8 } = 12$$

Solve for x

$$x = \frac{\log_{e} {(\frac{7}{6})} + 8}{4} \approx 2.03853767$$

Steps Using Definition of Logarithm

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

$$6e^{4x-8}+5=12$$

Subtract 5 from both sides of the equation.

$$6e^{4x-8}=7$$

Divide both sides by 6.

$$e^{4x-8}=\frac{7}{6}$$

Take the logarithm of both sides of the equation.

$$\log(e^{4x-8})=\log(\frac{7}{6})$$

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

$$\left(4x-8\right)\log(e)=\log(\frac{7}{6})$$

Divide both sides by \log(e).

$$4x-8=\frac{\log(\frac{7}{6})}{\log(e)}$$

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

$$4x-8=\log_{e}\left(\frac{7}{6}\right)$$

Add 8 to both sides of the equation.

$$4x=\ln(\frac{7}{6})-\left(-8\right)$$

Divide both sides by 4.

$$x=\frac{\ln(\frac{7}{6})+8}{4}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{2}+\frac{\ln(\frac{7}{6})+8}{4}$$

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