Solve a^x=bquada^2x=?

$$a ^ { x } = b \quad a ^ { 2 x } = ?$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=\frac{1}{a^{x}}\text{, }&x=0\text{ or }a\neq 0\\b\in \mathrm{C}\text{, }&a=0\text{ and }x\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ba^{2x}=a^{x}$$

The equation is in standard form.

$$a^{2x}b=a^{x}$$

Divide both sides by a^{2x}.

$$\frac{a^{2x}b}{a^{2x}}=\frac{a^{x}}{a^{2x}}$$

Dividing by a^{2x} undoes the multiplication by a^{2x}.

$$b=\frac{a^{x}}{a^{2x}}$$

Divide a^{x} by a^{2x}.

$$b=\frac{1}{a^{x}}$$

Solve for b

$$\left\{\begin{matrix}b=\frac{1}{a^{x}}\text{, }&a>0\text{ or }\left(Denominator(x)\text{bmod}2=1\text{ and }a<0\right)\\b\in \mathrm{R}\text{, }&a=0\text{ and }x>0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ba^{2x}=a^{x}$$

The equation is in standard form.

$$a^{2x}b=a^{x}$$

Divide both sides by a^{2x}.

$$\frac{a^{2x}b}{a^{2x}}=\frac{a^{x}}{a^{2x}}$$

Dividing by a^{2x} undoes the multiplication by a^{2x}.

$$b=\frac{a^{x}}{a^{2x}}$$

Divide a^{x} by a^{2x}.

$$b=\frac{1}{a^{x}}$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=e^{-\frac{2\pi n_{1}iRe(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}-\frac{2\pi n_{1}Im(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}+\frac{arg(\frac{1}{b})Im(x)+iarg(\frac{1}{b})Re(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}}\left(|b|\right)^{\frac{-Re(x)+iIm(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}}\text{, }n_{1}\in \mathrm{Z}\text{, }&b\neq 0\\a=0\text{, }&x\neq 0\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}a=0\text{, }&x>0\\a=\left(\frac{1}{b}\right)^{\frac{1}{x}}\text{, }&\left(Numerator(x)\text{bmod}2=1\text{ and }Denominator(x)\text{bmod}2=1\text{ and }\left(\frac{1}{b}\right)^{\frac{1}{x}}\neq 0\text{ and }b<0\right)\text{ or }\left(\left(\frac{1}{b}\right)^{\frac{1}{x}}>0\text{ and }x\neq 0\text{ and }b>0\right)\text{ or }\left(\left(\frac{1}{b}\right)^{\frac{1}{x}}<0\text{ and }x\neq 0\text{ and }Denominator(x)\text{bmod}2=1\text{ and }b>0\right)\\a=-\left(\frac{1}{b}\right)^{\frac{1}{x}}\text{, }&\left(b<0\text{ and }Numerator(x)\text{bmod}2=1\text{ and }Numerator(x)\text{bmod}2=0\text{ and }Denominator(x)\text{bmod}2=1\text{ and }\left(\frac{1}{b}\right)^{\frac{1}{x}}\neq 0\right)\text{ or }\left(b>0\text{ and }x\neq 0\text{ and }\left(\frac{1}{b}\right)^{\frac{1}{x}}<0\text{ and }Numerator(x)\text{bmod}2=0\right)\text{ or }\left(b>0\text{ and }x\neq 0\text{ and }\left(\frac{1}{b}\right)^{\frac{1}{x}}>0\text{ and }Numerator(x)\text{bmod}2=0\text{ and }Denominator(x)\text{bmod}2=1\right)\\a\neq 0\text{, }&b=1\text{ and }x=0\end{matrix}\right.$$