Solve log_{5}(5^2*5^3/sqrt{5})^2=

$$\log _ { 5 } ( \frac { 5 ^ { 2 } \cdot 5 ^ { 3 } } { \sqrt { 5 } } ) ^ { 2 } =$$

Evaluate

$$9$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$\log_{5}\left(\left(\frac{5^{5}}{\sqrt{5}}\right)^{2}\right)$$

Calculate 5 to the power of 5 and get 3125.

$$\log_{5}\left(\left(\frac{3125}{\sqrt{5}}\right)^{2}\right)$$

Rationalize the denominator of \frac{3125}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\log_{5}\left(\left(\frac{3125\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\right)^{2}\right)$$

The square of \sqrt{5} is 5.

$$\log_{5}\left(\left(\frac{3125\sqrt{5}}{5}\right)^{2}\right)$$

Divide 3125\sqrt{5} by 5 to get 625\sqrt{5}.

$$\log_{5}\left(\left(625\sqrt{5}\right)^{2}\right)$$

Expand \left(625\sqrt{5}\right)^{2}.

$$\log_{5}\left(625^{2}\left(\sqrt{5}\right)^{2}\right)$$

Calculate 625 to the power of 2 and get 390625.

$$\log_{5}\left(390625\left(\sqrt{5}\right)^{2}\right)$$

The square of \sqrt{5} is 5.

$$\log_{5}\left(390625\times 5\right)$$

Multiply 390625 and 5 to get 1953125.

$$\log_{5}\left(1953125\right)$$

The base 5 logarithm of 1953125 is 9.

$$9$$

Factor

$$3^{2}$$