Solve 128^x=0.25

$${ 128 }^{ x } =0.25$$

Solve for x

$$x=-\frac{2}{7}\approx -0.285714286$$

Steps Using Definition of Logarithm

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

$$128^{x}=0.25$$

Take the logarithm of both sides of the equation.

$$\log(128^{x})=\log(0.25)$$

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

$$x\log(128)=\log(0.25)$$

Divide both sides by \log(128).

$$x=\frac{\log(0.25)}{\log(128)}$$

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

$$x=\log_{128}\left(0.25\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{7\ln(2)}-\frac{2}{7}$$

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