Solve (27/8)^5*(27/8)^n=(3/2)^18

$$( \frac { 27 } { 8 } ) ^ { 5 } \times ( \frac { 27 } { 8 } ) ^ { n } = ( \frac { 3 } { 2 } ) ^ { 18 }$$

Solve for n

$$n=1$$

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

$$\frac{14348907}{32768}\times \left(\frac{27}{8}\right)^{n}=\frac{387420489}{262144}$$

Divide both sides of the equation by \frac{14348907}{32768}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\left(\frac{27}{8}\right)^{n}=\frac{27}{8}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{27}{8}\right)^{n})=\log(\frac{27}{8})$$

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

$$n\log(\frac{27}{8})=\log(\frac{27}{8})$$

Divide both sides by \log(\frac{27}{8}).

$$n=\frac{\log(\frac{27}{8})}{\log(\frac{27}{8})}$$

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

$$n=\log_{\frac{27}{8}}\left(\frac{27}{8}\right)$$