Solve frac{5^{m}*5^3*5^-2}{5^-3}=5^1

$$\frac { 5 ^ { m } \times 5 ^ { 3 } \times 5 ^ { - 2 } } { 5 ^ { - 3 } } = 5 ^ { 1 }$$

Solve for m

$$m=-3$$

Steps Using Definition of Logarithm

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

$$\frac{5^{m}\times 5^{1}}{5^{-3}}=5^{1}$$

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

$$5^{4}\times 5^{m}=5^{1}$$

Calculate 5 to the power of 1 and get 5.

$$5^{4}\times 5^{m}=5$$

Calculate 5 to the power of 4 and get 625.

$$625\times 5^{m}=5$$

Divide both sides by 625.

$$5^{m}=\frac{5}{625}$$

Reduce the fraction \frac{5}{625} to lowest terms by extracting and canceling out 5.

$$5^{m}=\frac{1}{125}$$

Take the logarithm of both sides of the equation.

$$\log(5^{m})=\log(\frac{1}{125})$$

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

$$m\log(5)=\log(\frac{1}{125})$$

Divide both sides by \log(5).

$$m=\frac{\log(\frac{1}{125})}{\log(5)}$$

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

$$m=\log_{5}\left(\frac{1}{125}\right)$$