Solve e^ax=Ce^bx

$$e ^ { a x } = C e ^ { b x }$$

Solve for C

$$C=e^{x\left(a-b\right)}$$

Steps for Solving Linear Equation

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

$$Ce^{bx}=e^{ax}$$

The equation is in standard form.

$$e^{bx}C=e^{ax}$$

Divide both sides by e^{bx}.

$$\frac{e^{bx}C}{e^{bx}}=\frac{e^{ax}}{e^{bx}}$$

Dividing by e^{bx} undoes the multiplication by e^{bx}.

$$C=\frac{e^{ax}}{e^{bx}}$$

Divide e^{ax} by e^{bx}.

$$C=e^{x\left(a-b\right)}$$

Solve for a

$$\left\{\begin{matrix}a=\frac{\ln(Ce^{bx})}{x}\text{, }&x\neq 0\text{ and }C>0\\a\in \mathrm{R}\text{, }&C=1\text{ and }x=0\end{matrix}\right.$$