Solve 3^7=3^3x

$$3 ^ { 7 } = 3 ^ { 3 x }$$

Solve for x

$$x = \frac{7}{3} = 2\frac{1}{3} \approx 2.333333333$$

Steps Using Definition of Logarithm

Calculate 3 to the power of 7 and get 2187.

$$2187=3^{3x}$$

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

$$3^{3x}=2187$$

Take the logarithm of both sides of the equation.

$$\log(3^{3x})=\log(2187)$$

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

$$3x\log(3)=\log(2187)$$

Divide both sides by \log(3).

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

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

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

Divide both sides by 3.

$$x=\frac{7}{3}$$

Solve for x (complex solution)

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

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