Solve x=x*4^-n

$$x = x \times 4 ^ { - n }$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }n=\frac{\pi n_{1}i}{\ln(2)}\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract x\times 4^{-n} from both sides.

$$x-x\times 4^{-n}=0$$

Reorder the terms.

$$-x\times 4^{-n}+x=0$$

Combine all terms containing x.

$$\left(-4^{-n}+1\right)x=0$$

The equation is in standard form.

$$\left(-\frac{1}{4^{n}}+1\right)x=0$$

Divide 0 by -4^{-n}+1.

$$x=0$$

Solve for n

$$\left\{\begin{matrix}\\n=0\text{, }&\text{unconditionally}\\n\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps Using Definition of Logarithm

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

$$x\times 4^{-n}=x$$

Divide both sides by x.

$$4^{-n}=1$$

Take the logarithm of both sides of the equation.

$$\log(4^{-n})=\log(1)$$

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

$$-n\log(4)=\log(1)$$

Divide both sides by \log(4).

$$-n=\frac{\log(1)}{\log(4)}$$

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

$$-n=\log_{4}\left(1\right)$$

Divide both sides by -1.

$$n=\frac{0}{-1}$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&n=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract x\times 4^{-n} from both sides.

$$x-x\times 4^{-n}=0$$

Reorder the terms.

$$-x\times 4^{-n}+x=0$$

Combine all terms containing x.

$$\left(-4^{-n}+1\right)x=0$$

The equation is in standard form.

$$\left(-\frac{1}{4^{n}}+1\right)x=0$$

Divide 0 by -4^{-n}+1.

$$x=0$$

Solve for n (complex solution)

$$\left\{\begin{matrix}\\n=-\frac{\pi n_{1}i}{\ln(2)}\text{, }n_{1}\in \mathrm{Z}\text{, }&\text{unconditionally}\\n\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$