Solve (33^7)^4/3^3=3^5*x

$$\frac { ( 33 ^ { 7 } ) ^ { 4 } } { 3 ^ { 3 } } = 3 ^ { 5 \cdot x }$$

Solve for x

$$x=\frac{28\log_{3}\left(11\right)}{5}+5\approx 17.222886696$$

Steps Using Definition of Logarithm

To raise a power to another power, multiply the exponents. Multiply 7 and 4 to get 28.

$$\frac{33^{28}}{3^{3}}=3^{5x}$$

Calculate 33 to the power of 28 and get 3299060778251569566188233498374847942355841.

$$\frac{3299060778251569566188233498374847942355841}{3^{3}}=3^{5x}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{3299060778251569566188233498374847942355841}{27}=3^{5x}$$

Divide 3299060778251569566188233498374847942355841 by 27 to get 122187436231539613562527166606475849716883.

$$122187436231539613562527166606475849716883=3^{5x}$$

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

$$3^{5x}=122187436231539613562527166606475849716883$$

Take the logarithm of both sides of the equation.

$$\log(3^{5x})=\log(122187436231539613562527166606475849716883)$$

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

$$5x\log(3)=\log(122187436231539613562527166606475849716883)$$

Divide both sides by \log(3).

$$5x=\frac{\log(122187436231539613562527166606475849716883)}{\log(3)}$$

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

$$5x=\log_{3}\left(122187436231539613562527166606475849716883\right)$$

Divide both sides by 5.

$$x=\frac{\log_{3}\left(122187436231539613562527166606475849716883\right)}{5}$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{5\ln(3)}+\frac{28\log_{3}\left(11\right)}{5}+5$$

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