Solve 3.2^x+3=12

$$3.2 ^ { x + 3 } = 12$$

Solve for x

$$x=-\log_{3.2}\left(\frac{1024}{375}\right)\approx -0.863641904$$

Steps Using Definition of Logarithm

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

$$3.2^{x+3}=12$$

Take the logarithm of both sides of the equation.

$$\log(3.2^{x+3})=\log(12)$$

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

$$\left(x+3\right)\log(3.2)=\log(12)$$

Divide both sides by \log(3.2).

$$x+3=\frac{\log(12)}{\log(3.2)}$$

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

$$x+3=\log_{3.2}\left(12\right)$$

Subtract 3 from both sides of the equation.

$$x=\frac{\ln(12)}{\ln(\frac{16}{5})}-3$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(3.2)}-\log_{3.2}\left(\frac{1024}{375}\right)$$

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