Solve 2^8=8^b

$$2 ^ { 8 } = 8 ^ { b }$$

Solve for b

$$b = \frac{8}{3} = 2\frac{2}{3} \approx 2.666666667$$

Steps Using Definition of Logarithm

Calculate 2 to the power of 8 and get 256.

$$256=8^{b}$$

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

$$8^{b}=256$$

Take the logarithm of both sides of the equation.

$$\log(8^{b})=\log(256)$$

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

$$b\log(8)=\log(256)$$

Divide both sides by \log(8).

$$b=\frac{\log(256)}{\log(8)}$$

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

$$b=\log_{8}\left(256\right)$$

Solve for b (complex solution)

$$b=\frac{2\pi n_{1}i}{3\ln(2)}+\frac{8}{3}$$

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