Solve 125*5^a=5^11

$$125 \times 5 ^ { a } = 5 ^ { 11 }$$

Solve for a

$$a=8$$

Steps Using Definition of Logarithm

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

$$125\times 5^{a}=48828125$$

Divide both sides by 125.

$$5^{a}=390625$$

Take the logarithm of both sides of the equation.

$$\log(5^{a})=\log(390625)$$

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

$$a\log(5)=\log(390625)$$

Divide both sides by \log(5).

$$a=\frac{\log(390625)}{\log(5)}$$

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

$$a=\log_{5}\left(390625\right)$$