Solve aq^b=a*q^5

$$a q ^ { b } = a \cdot q ^ { 5 }$$

Solve for a (complex solution)

$$\left\{\begin{matrix}\\a=0\text{, }&\text{unconditionally}\\a\in \mathrm{C}\text{, }&q=0\text{ or }q=1\text{ or }\exists n_{1}\in \mathrm{Z}\text{ : }b=\frac{2\pi n_{1}i}{\ln(q)}+5\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract aq^{5} from both sides.

$$aq^{b}-aq^{5}=0$$

Reorder the terms.

$$-aq^{5}+aq^{b}=0$$

Combine all terms containing a.

$$\left(-q^{5}+q^{b}\right)a=0$$

The equation is in standard form.

$$\left(q^{b}-q^{5}\right)a=0$$

Divide 0 by q^{b}-q^{5}.

$$a=0$$

Solve for a

$$\left\{\begin{matrix}a=0\text{, }&\left(q<0\text{ and }Denominator(b)\text{bmod}2=1\right)\text{ or }\left(q=0\text{ and }b>0\right)\text{ or }q>0\\a\in \mathrm{R}\text{, }&\left(q\neq 0\text{ and }b=5\right)\text{ or }q=1\text{ or }\left(q=0\text{ and }b>0\right)\text{ or }\left(q=-1\text{ and }Numerator(b)\text{bmod}2=1\text{ and }Denominator(b)\text{bmod}2=1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract aq^{5} from both sides.

$$aq^{b}-aq^{5}=0$$

Reorder the terms.

$$-aq^{5}+aq^{b}=0$$

Combine all terms containing a.

$$\left(-q^{5}+q^{b}\right)a=0$$

The equation is in standard form.

$$\left(q^{b}-q^{5}\right)a=0$$

Divide 0 by q^{b}-q^{5}.

$$a=0$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=\frac{2\pi n_{1}i}{\ln(q)}+5\text{, }n_{1}\in \mathrm{Z}\text{, }&q\neq 0\text{ and }q\neq 1\\b\in \mathrm{C}\text{, }&a=0\text{ or }q=0\text{ or }q=1\end{matrix}\right.$$

Solve for b

$$\left\{\begin{matrix}\\b=5\text{, }&\text{unconditionally}\\b\in \mathrm{R}\text{, }&\left(a=0\text{ and }q>0\right)\text{ or }\left(a=0\text{ and }q<0\text{ and }Denominator(b)\text{bmod}2=1\right)\text{ or }q=1\text{ or }\left(q=-1\text{ and }Numerator(b)\text{bmod}2=1\text{ and }Denominator(b)\text{bmod}2=1\right)\\b>0\text{, }&q=0\end{matrix}\right.$$