Solve 4^12-5x=2048

$$4 ^ { 12 - 5 x } = 2048$$

Solve for x

$$x = \frac{13}{10} = 1\frac{3}{10} = 1.3$$

Steps Using Definition of Logarithm

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

$$4^{-5x+12}=2048$$

Take the logarithm of both sides of the equation.

$$\log(4^{-5x+12})=\log(2048)$$

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

$$\left(-5x+12\right)\log(4)=\log(2048)$$

Divide both sides by \log(4).

$$-5x+12=\frac{\log(2048)}{\log(4)}$$

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

$$-5x+12=\log_{4}\left(2048\right)$$

Subtract 12 from both sides of the equation.

$$-5x=\frac{11}{2}-12$$

Divide both sides by -5.

$$x=-\frac{\frac{13}{2}}{-5}$$

Solve for x (complex solution)

$$x=-\frac{i\pi n_{1}}{5\ln(2)}+\frac{13}{10}$$

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