Solve 4^x+2=14.5

$${ 4 }^{ x+2 } =14.5$$

Solve for x

$$x=\frac{\log_{2}\left(29\right)-5}{2}\approx -0.071009502$$

Steps Using Definition of Logarithm

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

$$4^{x+2}=14.5$$

Take the logarithm of both sides of the equation.

$$\log(4^{x+2})=\log(14.5)$$

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

$$\left(x+2\right)\log(4)=\log(14.5)$$

Divide both sides by \log(4).

$$x+2=\frac{\log(14.5)}{\log(4)}$$

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

$$x+2=\log_{4}\left(14.5\right)$$

Subtract 2 from both sides of the equation.

$$x=\frac{\log_{2}\left(29\right)-1}{2}-2$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(2)}+\frac{\log_{2}\left(29\right)}{2}-2.5$$

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