Solve 625=1/25*5^n-2

$$625 = \frac { 1 } { 25 } \times 5 ^ { n - 2 }$$

Solve for n

$$n=8$$

Steps Using Definition of Logarithm

Multiply both sides by 25, the reciprocal of \frac{1}{25}.

$$625\times 25=5^{n-2}$$

Multiply 625 and 25 to get 15625.

$$15625=5^{n-2}$$

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

$$5^{n-2}=15625$$

Take the logarithm of both sides of the equation.

$$\log(5^{n-2})=\log(15625)$$

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

$$\left(n-2\right)\log(5)=\log(15625)$$

Divide both sides by \log(5).

$$n-2=\frac{\log(15625)}{\log(5)}$$

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

$$n-2=\log_{5}\left(15625\right)$$

Add 2 to both sides of the equation.

$$n=6-\left(-2\right)$$

Solve for n (complex solution)

$$n=\frac{2\pi n_{1}i}{\ln(5)}+8$$

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