Solve frac{sqrt{3}*sqrt[4]{9}}{3}=3^x

$$\frac { \sqrt { 3 } \cdot \sqrt[ 4 ] { 9 } } { 3 } = 3 ^ { x }$$

Solve for x

$$x=0$$

Steps Using Definition of Logarithm

Multiply both sides of the equation by 3.

$$\sqrt{3}\sqrt[4]{9}=3\times 3^{x}$$

Rewrite \sqrt[4]{9} as \sqrt[4]{3^{2}}. Convert from radical to exponential form and cancel out 2 in the exponent. Convert back to radical form.

$$\sqrt[4]{9}=\sqrt[4]{3^{2}}=3^{\frac{2}{4}}=3^{\frac{1}{2}}=\sqrt{3}$$

Insert the obtained value back in the expression.

$$\sqrt{3}\sqrt{3}=3\times 3^{x}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$3=3\times 3^{x}$$

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

$$3\times 3^{x}=3$$

Divide both sides by 3.

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

Divide 3 by 3 to get 1.

$$3^{x}=1$$

Take the logarithm of both sides of the equation.

$$\log(3^{x})=\log(1)$$

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

$$x\log(3)=\log(1)$$

Divide both sides by \log(3).

$$x=\frac{\log(1)}{\log(3)}$$

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

$$x=\log_{3}\left(1\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(3)}$$

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