Solve V5^6=V5^x

$$V 5 ^ { 6 } = V 5 ^ { x }$$

Solve for V (complex solution)

$$\left\{\begin{matrix}\\V=0\text{, }&\text{unconditionally}\\V\in \mathrm{C}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }x=\frac{2\pi n_{1}i}{\ln(5)}+6\end{matrix}\right.$$

Steps for Solving Linear Equation

Calculate 5 to the power of 6 and get 15625.

$$V\times 15625=V\times 5^{x}$$

Subtract V\times 5^{x} from both sides.

$$V\times 15625-V\times 5^{x}=0$$

Reorder the terms.

$$-V\times 5^{x}+15625V=0$$

Combine all terms containing V.

$$\left(-5^{x}+15625\right)V=0$$

The equation is in standard form.

$$\left(15625-5^{x}\right)V=0$$

Divide 0 by 15625-5^{x}.

$$V=0$$

Solve for V

$$\left\{\begin{matrix}\\V=0\text{, }&\text{unconditionally}\\V\in \mathrm{R}\text{, }&x=6\end{matrix}\right.$$

Steps for Solving Linear Equation

Calculate 5 to the power of 6 and get 15625.

$$V\times 15625=V\times 5^{x}$$

Subtract V\times 5^{x} from both sides.

$$V\times 15625-V\times 5^{x}=0$$

Reorder the terms.

$$-V\times 5^{x}+15625V=0$$

Combine all terms containing V.

$$\left(-5^{x}+15625\right)V=0$$

The equation is in standard form.

$$\left(15625-5^{x}\right)V=0$$

Divide 0 by 15625-5^{x}.

$$V=0$$

Solve for x

$$\left\{\begin{matrix}\\x=6\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&V=0\end{matrix}\right.$$

Steps Using Definition of Logarithm

Calculate 5 to the power of 6 and get 15625.

$$V\times 15625=V\times 5^{x}$$

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

$$V\times 5^{x}=V\times 15625$$

Use the rules of exponents and logarithms to solve the equation.

$$V\times 5^{x}=15625V$$

Divide both sides by V.

$$5^{x}=15625$$

Take the logarithm of both sides of the equation.

$$\log(5^{x})=\log(15625)$$

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

$$x\log(5)=\log(15625)$$

Divide both sides by \log(5).

$$x=\frac{\log(15625)}{\log(5)}$$

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

$$x=\log_{5}\left(15625\right)$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=\frac{2\pi n_{1}i}{\ln(5)}+6\text{, }n_{1}\in \mathrm{Z}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&V=0\end{matrix}\right.$$