Solve 3^2*27^2x=3^4

$$3 ^ { 2 } \cdot 27 ^ { 2 x } = 3 ^ { 4 }$$

Solve for x

$$x=\frac{1}{3}\approx 0.333333333$$

Steps Using Definition of Logarithm

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

$$9\times 27^{2x}=81$$

Divide both sides by 9.

$$27^{2x}=9$$

Take the logarithm of both sides of the equation.

$$\log(27^{2x})=\log(9)$$

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

$$2x\log(27)=\log(9)$$

Divide both sides by \log(27).

$$2x=\frac{\log(9)}{\log(27)}$$

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

$$2x=\log_{27}\left(9\right)$$

Divide both sides by 2.

$$x=\frac{\frac{2}{3}}{2}$$

Solve for x (complex solution)

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

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