Solve 0.3=4^x-3

$$0.3= { 4 }^{ x-3 }$$

Solve for x

$$x = \frac{\log_{2} {(0.6)} + 5}{2} \approx 2.131517203$$

Steps Using Definition of Logarithm

Swap sides so that all variable terms are on the left hand side.

$$4^{x-3}=0.3$$

Take the logarithm of both sides of the equation.

$$\log(4^{x-3})=\log(0.3)$$

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

$$\left(x-3\right)\log(4)=\log(0.3)$$

Divide both sides by \log(4).

$$x-3=\frac{\log(0.3)}{\log(4)}$$

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

$$x-3=\log_{4}\left(0.3\right)$$

Add 3 to both sides of the equation.

$$x=\frac{\ln(\frac{3}{10})}{2\ln(2)}-\left(-3\right)$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(2)}+\frac{\log_{2}\left(0.6\right)}{2}+2.5$$

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