Solve 4^x-32.25=14

$${ 4 }^{ x } -32.25=14$$

Solve for x

$$x = \frac{\log_{2} {(185)} - 2}{2} \approx 2.76569073$$

Steps Using Definition of Logarithm

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

$$4^{x}-32.25=14$$

Add 32.25 to both sides of the equation.

$$4^{x}=46.25$$

Take the logarithm of both sides of the equation.

$$\log(4^{x})=\log(46.25)$$

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

$$x\log(4)=\log(46.25)$$

Divide both sides by \log(4).

$$x=\frac{\log(46.25)}{\log(4)}$$

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

$$x=\log_{4}\left(46.25\right)$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(2)}+\frac{\log_{2}\left(185\right)}{2}-1$$

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