Solve {left(8/9right)}^x=81/64

$${ \left( \frac{ 8 }{ 9 } \right) }^{ x } = \frac{ 81 }{ 64 }$$

Solve for x

$$x=-2$$

Steps Using Definition of Logarithm

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

$$\left(\frac{8}{9}\right)^{x}=\frac{81}{64}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{8}{9}\right)^{x})=\log(\frac{81}{64})$$

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

$$x\log(\frac{8}{9})=\log(\frac{81}{64})$$

Divide both sides by \log(\frac{8}{9}).

$$x=\frac{\log(\frac{81}{64})}{\log(\frac{8}{9})}$$

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

$$x=\log_{\frac{8}{9}}\left(\frac{81}{64}\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(\frac{8}{9})}+2\log_{\frac{8}{9}}\left(\frac{9}{8}\right)$$

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