Solve 4^x^+16=17.2

$${ 4 }^{ { x }^{ } } +16=17.2$$

Solve for x

$$x=\frac{\log_{2}\left(1.2\right)}{2}\approx 0.131517203$$

Steps Using Definition of Logarithm

Calculate x to the power of 1 and get x.

$$4^{x}+16=17.2$$

Subtract 17.2 from both sides.

$$4^{x}+16-17.2=0$$

Subtract 17.2 from 16 to get -1.2.

$$4^{x}-1.2=0$$

Add 1.2 to both sides of the equation.

$$4^{x}=1.2$$

Take the logarithm of both sides of the equation.

$$\log(4^{x})=\log(1.2)$$

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

$$x\log(4)=\log(1.2)$$

Divide both sides by \log(4).

$$x=\frac{\log(1.2)}{\log(4)}$$

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

$$x=\log_{4}\left(1.2\right)$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(2)}+\frac{\log_{2}\left(1.2\right)}{2}$$

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