Solve 5^3y*5^7=5^19

$$5 ^ { 3 y } \times 5 ^ { 7 } = 5 ^ { 19 }$$

Solve for y

$$y=4$$

Steps Using Definition of Logarithm

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

$$78125\times 5^{3y}=19073486328125$$

Divide both sides by 78125.

$$5^{3y}=244140625$$

Take the logarithm of both sides of the equation.

$$\log(5^{3y})=\log(244140625)$$

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

$$3y\log(5)=\log(244140625)$$

Divide both sides by \log(5).

$$3y=\frac{\log(244140625)}{\log(5)}$$

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

$$3y=\log_{5}\left(244140625\right)$$

Divide both sides by 3.

$$y=\frac{12}{3}$$

Solve for y (complex solution)

$$y=\frac{2\pi n_{1}i}{3\ln(5)}+4$$

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