Solve frac{5^{m}*5^3*5^-2}{5^-5}=5^12

$$\frac { 5 ^ { m } \times 5 ^ { 3 } \times 5 ^ { - 2 } } { 5 ^ { - 5 } } = 5 ^ { 12 }$$

Solve for m

$$m=6$$

Steps Using Definition of Logarithm

To multiply powers of the same base, add their exponents. Add 3 and -2 to get 1.

$$\frac{5^{m}\times 5^{1}}{5^{-5}}=5^{12}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$5^{6}\times 5^{m}=5^{12}$$

Calculate 5 to the power of 12 and get 244140625.

$$5^{6}\times 5^{m}=244140625$$

Calculate 5 to the power of 6 and get 15625.

$$15625\times 5^{m}=244140625$$

Divide both sides by 15625.

$$5^{m}=\frac{244140625}{15625}$$

Divide 244140625 by 15625 to get 15625.

$$5^{m}=15625$$

Take the logarithm of both sides of the equation.

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

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

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

Divide both sides by \log(5).

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

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

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