Solve 3+9^(2/5)x=sqrt[3]{216}

$$3 + 9 ^ { ( 2 / 5 ) x } = \sqrt[ 3 ] { 216 }$$

Solve for x

$$x = \frac{5}{4} = 1\frac{1}{4} = 1.25$$

Steps Using Definition of Logarithm

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

$$9^{\frac{2}{5}x}+3=\sqrt[3]{216}$$

Subtract 3 from both sides of the equation.

$$9^{\frac{2}{5}x}=3$$

Take the logarithm of both sides of the equation.

$$\log(9^{\frac{2}{5}x})=\log(3)$$

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

$$\frac{2}{5}x\log(9)=\log(3)$$

Divide both sides by \log(9).

$$\frac{2}{5}x=\frac{\log(3)}{\log(9)}$$

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

$$\frac{2}{5}x=\log_{9}\left(3\right)$$

Divide both sides of the equation by \frac{2}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{\frac{1}{2}}{\frac{2}{5}}$$

Solve for x (complex solution)

$$x=\frac{i\times 5\pi n_{1}}{2\ln(3)}+\frac{5}{4}$$

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