Solve 5^10div5^8=(1/5)^n

$$5 ^ { 10 } \div 5 ^ { 8 } = ( \frac { 1 } { 5 } ) ^ { n }$$

Solve for n

$$n=-2$$

Steps Using Definition of Logarithm

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 8 from 10 to get 2.

$$5^{2}=\left(\frac{1}{5}\right)^{n}$$

Calculate 5 to the power of 2 and get 25.

$$25=\left(\frac{1}{5}\right)^{n}$$

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

$$\left(\frac{1}{5}\right)^{n}=25$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{5}\right)^{n})=\log(25)$$

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

$$n\log(\frac{1}{5})=\log(25)$$

Divide both sides by \log(\frac{1}{5}).

$$n=\frac{\log(25)}{\log(\frac{1}{5})}$$

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

$$n=\log_{\frac{1}{5}}\left(25\right)$$