Solve 3^-7x-2=0

$${ 3 }^{ -7x } -2=0$$

Solve for x

$$x=-\frac{\log_{3}\left(2\right)}{7}\approx -0.090132822$$

Steps Using Definition of Logarithm

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

$$3^{-7x}-2=0$$

Add 2 to both sides of the equation.

$$3^{-7x}=2$$

Take the logarithm of both sides of the equation.

$$\log(3^{-7x})=\log(2)$$

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

$$-7x\log(3)=\log(2)$$

Divide both sides by \log(3).

$$-7x=\frac{\log(2)}{\log(3)}$$

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

$$-7x=\log_{3}\left(2\right)$$

Divide both sides by -7.

$$x=\frac{\log_{3}\left(2\right)}{-7}$$

Solve for x (complex solution)

$$x=-\frac{2\pi n_{1}i}{7\ln(3)}-\frac{\log_{3}\left(2\right)}{7}$$

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