Solve 729^4-x=3

$$729 ^ { 4 - x } = 3$$

Solve for x

$$x = \frac{23}{6} = 3\frac{5}{6} \approx 3.833333333$$

Steps Using Definition of Logarithm

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

$$729^{-x+4}=3$$

Take the logarithm of both sides of the equation.

$$\log(729^{-x+4})=\log(3)$$

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

$$\left(-x+4\right)\log(729)=\log(3)$$

Divide both sides by \log(729).

$$-x+4=\frac{\log(3)}{\log(729)}$$

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

$$-x+4=\log_{729}\left(3\right)$$

Subtract 4 from both sides of the equation.

$$-x=\frac{1}{6}-4$$

Divide both sides by -1.

$$x=-\frac{\frac{23}{6}}{-1}$$

Solve for x (complex solution)

$$x=-\frac{i\pi n_{1}}{3\ln(3)}+\frac{23}{6}$$

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