Solve 5^m+5^-3=5^5

$$5 ^ { m } + 5 ^ { - 3 } = 5 ^ { 5 }$$

Solve for m

$$m=\log_{5}\left(390624\right)-3\approx 4.999998409$$

Steps Using Definition of Logarithm

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

$$5^{m}+\frac{1}{125}=3125$$

Subtract \frac{1}{125} from both sides of the equation.

$$5^{m}=\frac{390624}{125}$$

Take the logarithm of both sides of the equation.

$$\log(5^{m})=\log(\frac{390624}{125})$$

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

$$m\log(5)=\log(\frac{390624}{125})$$

Divide both sides by \log(5).

$$m=\frac{\log(\frac{390624}{125})}{\log(5)}$$

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

$$m=\log_{5}\left(\frac{390624}{125}\right)$$

Solve for m (complex solution)

$$m=\frac{i\times 2\pi n_{1}}{\ln(5)}+\log_{5}\left(390624\right)-3$$

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