Solve [(5^{4})^{2}*5^{3^2}]:(5^3)^2=5^a

$$[ ( 5 ^ { 4 } ) ^ { 2 } \cdot 5 ^ { 3 ^ { 2 } } ] : ( 5 ^ { 3 } ) ^ { 2 } = 5 ^ { a }$$

Solve for a

$$a=11$$

Steps Using Definition of Logarithm

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\frac{5^{8}\times 5^{3^{2}}}{\left(5^{3}\right)^{2}}=5^{a}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{5^{8}\times 5^{3^{2}}}{5^{6}}=5^{a}$$

Cancel out 5^{6} in both numerator and denominator.

$$5^{2}\times 5^{3^{2}}=5^{a}$$

Calculate 5 to the power of 2 and get 25.

$$25\times 5^{3^{2}}=5^{a}$$

Calculate 3 to the power of 2 and get 9.

$$25\times 5^{9}=5^{a}$$

Calculate 5 to the power of 9 and get 1953125.

$$25\times 1953125=5^{a}$$

Multiply 25 and 1953125 to get 48828125.

$$48828125=5^{a}$$

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

$$5^{a}=48828125$$

Take the logarithm of both sides of the equation.

$$\log(5^{a})=\log(48828125)$$

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

$$a\log(5)=\log(48828125)$$

Divide both sides by \log(5).

$$a=\frac{\log(48828125)}{\log(5)}$$

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

$$a=\log_{5}\left(48828125\right)$$

Solve for a (complex solution)

$$a=\frac{2\pi n_{1}i}{\ln(5)}+11$$

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