Solve 992=8^x

$$992 = 8 ^ { x }$$

Solve for x

$$x = \frac{\log_{2} {(31)} + 5}{3} \approx 3.318065437$$

Steps Using Definition of Logarithm

Swap sides so that all variable terms are on the left hand side.

$$8^{x}=992$$

Take the logarithm of both sides of the equation.

$$\log(8^{x})=\log(992)$$

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

$$x\log(8)=\log(992)$$

Divide both sides by \log(8).

$$x=\frac{\log(992)}{\log(8)}$$

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

$$x=\log_{8}\left(992\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{3\ln(2)}+\frac{\log_{2}\left(992\right)}{3}$$

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