Solve 5^6=5^x*5^3

$$5 ^ { 6 } = 5 ^ { x } \cdot 5 ^ { 3 }$$

Solve for x

$$x=3$$

Steps Using Definition of Logarithm

Calculate 5 to the power of 6 and get 15625.

$$15625=5^{x}\times 5^{3}$$

Calculate 5 to the power of 3 and get 125.

$$15625=5^{x}\times 125$$

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

$$5^{x}\times 125=15625$$

Divide both sides by 125.

$$5^{x}=\frac{15625}{125}$$

Divide 15625 by 125 to get 125.

$$5^{x}=125$$

Take the logarithm of both sides of the equation.

$$\log(5^{x})=\log(125)$$

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

$$x\log(5)=\log(125)$$

Divide both sides by \log(5).

$$x=\frac{\log(125)}{\log(5)}$$

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

$$x=\log_{5}\left(125\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(5)}+3$$

$$n_{1}\in \mathrm{Z}$$