Solve (25/16)^-frac{3{2}}=2^p*5^8

$$( \frac { 25 } { 16 } ) ^ { - \frac { 3 } { 2 } } = 2 ^ { p } \times 5 ^ { 8 }$$

Solve for p

$$p=-11\log_{2}\left(5\right)+6\approx -19.541209044$$

Steps Using Definition of Logarithm

Calculate \frac{25}{16} to the power of -\frac{3}{2} and get \frac{64}{125}.

$$\frac{64}{125}=2^{p}\times 5^{8}$$

Calculate 5 to the power of 8 and get 390625.

$$\frac{64}{125}=2^{p}\times 390625$$

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

$$2^{p}\times 390625=\frac{64}{125}$$

Divide both sides by 390625.

$$2^{p}=\frac{\frac{64}{125}}{390625}$$

Express \frac{\frac{64}{125}}{390625} as a single fraction.

$$2^{p}=\frac{64}{125\times 390625}$$

Multiply 125 and 390625 to get 48828125.

$$2^{p}=\frac{64}{48828125}$$

Take the logarithm of both sides of the equation.

$$\log(2^{p})=\log(\frac{64}{48828125})$$

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

$$p\log(2)=\log(\frac{64}{48828125})$$

Divide both sides by \log(2).

$$p=\frac{\log(\frac{64}{48828125})}{\log(2)}$$

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

$$p=\log_{2}\left(\frac{64}{48828125}\right)$$