Solve 1/9=243*({left(1/3right)}^x)

$$\frac{ 1 }{ 9 } =243 \times ( { \left( \frac{ 1 }{ 3 } \right) }^{ x } )$$

Solve for x

$$x=7$$

Steps Using Definition of Logarithm

Divide both sides by 243.

$$\frac{\frac{1}{9}}{243}=\left(\frac{1}{3}\right)^{x}$$

Express \frac{\frac{1}{9}}{243} as a single fraction.

$$\frac{1}{9\times 243}=\left(\frac{1}{3}\right)^{x}$$

Multiply 9 and 243 to get 2187.

$$\frac{1}{2187}=\left(\frac{1}{3}\right)^{x}$$

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

$$\left(\frac{1}{3}\right)^{x}=\frac{1}{2187}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{3}\right)^{x})=\log(\frac{1}{2187})$$

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

$$x\log(\frac{1}{3})=\log(\frac{1}{2187})$$

Divide both sides by \log(\frac{1}{3}).

$$x=\frac{\log(\frac{1}{2187})}{\log(\frac{1}{3})}$$

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

$$x=\log_{\frac{1}{3}}\left(\frac{1}{2187}\right)$$

Solve for x (complex solution)

$$x=-\frac{2\pi n_{1}i}{\ln(3)}+7$$

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