Solve 32^x+3=2

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

Solve for x

$$x = -\frac{14}{5} = -2\frac{4}{5} = -2.8$$

Steps Using Definition of Logarithm

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

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

Take the logarithm of both sides of the equation.

$$\log(32^{x+3})=\log(2)$$

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

$$\left(x+3\right)\log(32)=\log(2)$$

Divide both sides by \log(32).

$$x+3=\frac{\log(2)}{\log(32)}$$

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

$$x+3=\log_{32}\left(2\right)$$

Subtract 3 from both sides of the equation.

$$x=\frac{1}{5}-3$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{5\ln(2)}-\frac{14}{5}$$

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