Solve 3^n/32div8/9=0.889

$$\frac { 3 ^ { n } } { 32 } \div \frac { 8 } { 9 } = 0.889$$

Solve for n

$$n=\log_{3}\left(\frac{28448}{1125}\right)\approx 2.940341062$$

Steps Using Definition of Logarithm

Multiply both sides by \frac{8}{9}.

$$\frac{3^{n}}{32}=0.889\times \frac{8}{9}$$

Multiply both sides of the equation by 288, the least common multiple of 32,9.

$$9\times 3^{n}=256.032\times \frac{8}{9}$$

Multiply 256.032 and \frac{8}{9} to get \frac{28448}{125}.

$$9\times 3^{n}=\frac{28448}{125}$$

Divide both sides by 9.

$$3^{n}=\frac{\frac{28448}{125}}{9}$$

Express \frac{\frac{28448}{125}}{9} as a single fraction.

$$3^{n}=\frac{28448}{125\times 9}$$

Multiply 125 and 9 to get 1125.

$$3^{n}=\frac{28448}{1125}$$

Take the logarithm of both sides of the equation.

$$\log(3^{n})=\log(\frac{28448}{1125})$$

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

$$n\log(3)=\log(\frac{28448}{1125})$$

Divide both sides by \log(3).

$$n=\frac{\log(\frac{28448}{1125})}{\log(3)}$$

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

$$n=\log_{3}\left(\frac{28448}{1125}\right)$$