Solve 4^2=32^3x

$$4 ^ { 2 } = 32 ^ { 3 x }$$

Solve for x

$$x=\frac{4}{15}\approx 0.266666667$$

Steps Using Definition of Logarithm

Calculate 4 to the power of 2 and get 16.

$$16=32^{3x}$$

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

$$32^{3x}=16$$

Take the logarithm of both sides of the equation.

$$\log(32^{3x})=\log(16)$$

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

$$3x\log(32)=\log(16)$$

Divide both sides by \log(32).

$$3x=\frac{\log(16)}{\log(32)}$$

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

$$3x=\log_{32}\left(16\right)$$

Divide both sides by 3.

$$x=\frac{\frac{4}{5}}{3}$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{15\ln(2)}+\frac{4}{15}$$

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