Solve {3}^frac{3-x{5}}=4

$${ 3 }^{ \frac{ 3-x }{ 5 } } = 4$$

Solve for x

$$x=-10\log_{3}\left(2\right)+3\approx -3.309297536$$

Steps Using Definition of Logarithm

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

$$3^{\frac{1}{5}\left(-x+3\right)}=4$$

Take the logarithm of both sides of the equation.

$$\log(3^{\frac{1}{5}\left(-x+3\right)})=\log(4)$$

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

$$\frac{1}{5}\left(-x+3\right)\log(3)=\log(4)$$

Divide both sides by \log(3).

$$\frac{1}{5}\left(-x+3\right)=\frac{\log(4)}{\log(3)}$$

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

$$\frac{1}{5}\left(-x+3\right)=\log_{3}\left(4\right)$$

Multiply both sides by 5.

$$-x+3=\frac{2\log_{3}\left(2\right)}{\frac{1}{5}}$$

Subtract 3 from both sides of the equation.

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

Divide both sides by -1.

$$x=\frac{10\log_{3}\left(2\right)-3}{-1}$$

Solve for x (complex solution)

$$x=-\frac{10\pi n_{1}i}{\ln(3)}-10\log_{3}\left(2\right)+3$$

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