Solve (2/9)^3*(2/9)^8=(2/9)^2x

$$( \frac { 2 } { 9 } ) ^ { 3 } \times ( \frac { 2 } { 9 } ) ^ { 8 } = ( \frac { 2 } { 9 } ) ^ { 2 x }$$

Solve for x

$$x = \frac{11}{2} = 5\frac{1}{2} = 5.5$$

Steps Using Definition of Logarithm

To multiply powers of the same base, add their exponents. Add 3 and 8 to get 11.

$$\left(\frac{2}{9}\right)^{11}=\left(\frac{2}{9}\right)^{2x}$$

Calculate \frac{2}{9} to the power of 11 and get \frac{2048}{31381059609}.

$$\frac{2048}{31381059609}=\left(\frac{2}{9}\right)^{2x}$$

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

$$\left(\frac{2}{9}\right)^{2x}=\frac{2048}{31381059609}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{2}{9}\right)^{2x})=\log(\frac{2048}{31381059609})$$

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

$$2x\log(\frac{2}{9})=\log(\frac{2048}{31381059609})$$

Divide both sides by \log(\frac{2}{9}).

$$2x=\frac{\log(\frac{2048}{31381059609})}{\log(\frac{2}{9})}$$

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

$$2x=\log_{\frac{2}{9}}\left(\frac{2048}{31381059609}\right)$$

Divide both sides by 2.

$$x=\frac{11}{2}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(\frac{2}{9})}-\frac{\log_{\frac{2}{9}}\left(\frac{31381059609}{2048}\right)}{2}$$

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