Solve for b (5^6)/(5^2)=a^b

$\frac{5^{6}}{5^{2}} = a^{b}$

Rewrite the equation as $a^{b} = \frac{5^{6}}{5^{2}}$ .

$a^{b} = \frac{5^{6}}{5^{2}}$

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (a^{b}) = \ln (\frac{5^{6}}{5^{2}})$

Expand $\ln (a^{b})$ by moving $b$ outside the logarithm.

$b \ln (a) = \ln (\frac{5^{6}}{5^{2}})$

Simplify the right side.

Simplify $\ln (\frac{5^{6}}{5^{2}})$ .

Cancel the common factor of $5^{6}$ and $5^{2}$ .

Factor $5^{2}$ out of $5^{6}$ .

$b \ln (a) = \ln (\frac{5^{2} \cdot 5^{4}}{5^{2}})$

Cancel the common factors.

Multiply by $1$ .

$b \ln (a) = \ln (\frac{5^{2} \cdot 5^{4}}{5^{2} \cdot 1})$

Cancel the common factor.

$b \ln (a) = \ln (\frac{5^{2} \cdot 5^{4}}{5^{2} \cdot 1})$

Rewrite the expression.

$b \ln (a) = \ln (\frac{5^{4}}{1})$

Divide $5^{4}$ by $1$ .

$b \ln (a) = \ln (5^{4})$

Raise $5$ to the power of $4$ .

$b \ln (a) = \ln (625)$

Divide each term in $b \ln (a) = \ln (625)$ by $\ln (a)$ and simplify.

Divide each term in $b \ln (a) = \ln (625)$ by $\ln (a)$ .

$\frac{b \ln (a)}{\ln (a)} = \frac{\ln (625)}{\ln (a)}$

Simplify the left side.

Cancel the common factor of $\ln (a)$ .

Cancel the common factor.

$\frac{b \ln (a)}{\ln (a)} = \frac{\ln (625)}{\ln (a)}$

Divide $b$ by $1$ .

$b = \frac{\ln (625)}{\ln (a)}$