Solve {left(125/8right)}^5*{left(125/8right)}^x={left(5/2right)}^18

$${ \left( \frac{ 125 }{ 8 } \right) }^{ 5 } \times { \left( \frac{ 125 }{ 8 } \right) }^{ x } = { \left( \frac{ 5 }{ 2 } \right) }^{ 18 }$$

Solve for x

$$x=1$$

Steps Using Definition of Logarithm

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

$$\frac{30517578125}{32768}\times \left(\frac{125}{8}\right)^{x}=\frac{3814697265625}{262144}$$

Divide both sides of the equation by \frac{30517578125}{32768}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\left(\frac{125}{8}\right)^{x}=\frac{125}{8}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{125}{8}\right)^{x})=\log(\frac{125}{8})$$

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

$$x\log(\frac{125}{8})=\log(\frac{125}{8})$$

Divide both sides by \log(\frac{125}{8}).

$$x=\frac{\log(\frac{125}{8})}{\log(\frac{125}{8})}$$

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

$$x=\log_{\frac{125}{8}}\left(\frac{125}{8}\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{3\ln(\frac{5}{2})}+\frac{\log_{\frac{5}{2}}\left(\frac{125}{8}\right)}{3}$$

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