Solve 81^3*6^2*18^-12=3^12-x

$${ 81 }^{ 3 } \times { 6 }^{ 2 } \times { 18 }^{ -12 } = { 3 }^{ 12-x }$$

Solve for x

$$x=10\log_{3}\left(2\right)+22\approx 28.309297536$$

Steps Using Definition of Logarithm

Calculate 81 to the power of 3 and get 531441.

$$531441\times 6^{2}\times 18^{-12}=3^{12-x}$$

Calculate 6 to the power of 2 and get 36.

$$531441\times 36\times 18^{-12}=3^{12-x}$$

Multiply 531441 and 36 to get 19131876.

$$19131876\times 18^{-12}=3^{12-x}$$

Calculate 18 to the power of -12 and get \frac{1}{1156831381426176}.

$$19131876\times \frac{1}{1156831381426176}=3^{12-x}$$

Multiply 19131876 and \frac{1}{1156831381426176} to get \frac{1}{60466176}.

$$\frac{1}{60466176}=3^{12-x}$$

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

$$3^{12-x}=\frac{1}{60466176}$$

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

$$3^{-x+12}=\frac{1}{60466176}$$

Take the logarithm of both sides of the equation.

$$\log(3^{-x+12})=\log(\frac{1}{60466176})$$

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

$$\left(-x+12\right)\log(3)=\log(\frac{1}{60466176})$$

Divide both sides by \log(3).

$$-x+12=\frac{\log(\frac{1}{60466176})}{\log(3)}$$

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

$$-x+12=\log_{3}\left(\frac{1}{60466176}\right)$$

Subtract 12 from both sides of the equation.

$$-x=-10\log_{3}\left(6\right)-12$$

Divide both sides by -1.

$$x=\frac{-10\log_{3}\left(6\right)-12}{-1}$$

Solve for x (complex solution)

$$x=-\frac{2\pi n_{1}i}{\ln(3)}+10\log_{3}\left(2\right)+22$$

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