Solve 1/243=3^x

$$1 / 243 = 3 ^ { x }$$

Solve for x

$$x=-5$$

Steps Using Definition of Logarithm

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

$$3^{x}=\frac{1}{243}$$

Take the logarithm of both sides of the equation.

$$\log(3^{x})=\log(\frac{1}{243})$$

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

$$x\log(3)=\log(\frac{1}{243})$$

Divide both sides by \log(3).

$$x=\frac{\log(\frac{1}{243})}{\log(3)}$$

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

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

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(3)}-5$$

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