Solve 6e^-4x-10-4=63

$$6 e ^ { - 4 x - 10 } - 4 = 63$$

Solve for x

$$x=\frac{\ln(\frac{6}{67})-10}{4}\approx -3.103233288$$

Steps Using Definition of Logarithm

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

$$6e^{-4x-10}-4=63$$

Add 4 to both sides of the equation.

$$6e^{-4x-10}=67$$

Divide both sides by 6.

$$e^{-4x-10}=\frac{67}{6}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-4x-10})=\log(\frac{67}{6})$$

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

$$\left(-4x-10\right)\log(e)=\log(\frac{67}{6})$$

Divide both sides by \log(e).

$$-4x-10=\frac{\log(\frac{67}{6})}{\log(e)}$$

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

$$-4x-10=\log_{e}\left(\frac{67}{6}\right)$$

Add 10 to both sides of the equation.

$$-4x=\ln(\frac{67}{6})-\left(-10\right)$$

Divide both sides by -4.

$$x=\frac{\ln(\frac{67}{6})+10}{-4}$$

Solve for x (complex solution)

$$x=-\frac{\pi n_{1}i}{2}+\frac{\ln(\frac{6}{67})-10}{4}$$

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