Solve 3y=ye^0.1x

$$3y = y { e }^{ 0.1x }$$

Solve for x

$$\left\{\begin{matrix}\\x=10\ln(3)\approx 10.986122887\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&y=0\end{matrix}\right.$$

Steps Using Definition of Logarithm

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

$$ye^{0.1x}=3y$$

Divide both sides by y.

$$e^{0.1x}=3$$

Take the logarithm of both sides of the equation.

$$\log(e^{0.1x})=\log(3)$$

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

$$0.1x\log(e)=\log(3)$$

Divide both sides by \log(e).

$$0.1x=\frac{\log(3)}{\log(e)}$$

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

$$0.1x=\log_{e}\left(3\right)$$

Multiply both sides by 10.

$$x=\frac{\ln(3)}{0.1}$$

Solve for y

$$\left\{\begin{matrix}\\y=0\text{, }&\text{unconditionally}\\y\in \mathrm{R}\text{, }&x=10\ln(3)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract ye^{0.1x} from both sides.

$$3y-ye^{0.1x}=0$$

Reorder the terms.

$$-ye^{0.1x}+3y=0$$

Combine all terms containing y.

$$\left(-e^{0.1x}+3\right)y=0$$

The equation is in standard form.

$$\left(3-e^{\frac{x}{10}}\right)y=0$$

Divide 0 by 3-e^{0.1x}.

$$y=0$$