Solve 249^-x=7

$$249 ^ { - x } = 7$$

Solve for x

$$x=-\log_{249}\left(7\right)\approx -0.352682694$$

Steps Using Definition of Logarithm

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

$$249^{-x}=7$$

Take the logarithm of both sides of the equation.

$$\log(249^{-x})=\log(7)$$

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

$$-x\log(249)=\log(7)$$

Divide both sides by \log(249).

$$-x=\frac{\log(7)}{\log(249)}$$

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

$$-x=\log_{249}\left(7\right)$$

Divide both sides by -1.

$$x=\frac{\log_{249}\left(7\right)}{-1}$$

Solve for x (complex solution)

$$x=-\frac{2\pi n_{1}i}{\ln(249)}-\log_{249}\left(7\right)$$

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