Solve for x 10^((2x)/3)=15

$10^{\frac{2 x}{3}} = 15$

Take the base $10$ logarithm of both sides of the equation to remove the variable from the exponent.

$\log (10^{\frac{2 x}{3}}) = \log (15)$

Expand the left side.

Expand $\log (10^{\frac{2 x}{3}})$ by moving $\frac{2 x}{3}$ outside the logarithm.

$\frac{2 x}{3} \log (10) = \log (15)$

Logarithm base $10$ of $10$ is $1$ .

$\frac{2 x}{3} \cdot 1 = \log (15)$

Multiply $\frac{2 x}{3}$ by $1$ .

$\frac{2 x}{3} = \log (15)$

Multiply both sides of the equation by $\frac{3}{2}$ .

$\frac{3}{2} \cdot \frac{2 x}{3} = \frac{3}{2} \log (15)$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{3}{2} \cdot \frac{2 x}{3}$ .

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3}{2} \cdot \frac{2 x}{3} = \frac{3}{2} \log (15)$

Rewrite the expression.

$\frac{1}{2} (2 x) = \frac{3}{2} \log (15)$

Cancel the common factor of $2$ .

Factor $2$ out of $2 x$ .

$\frac{1}{2} (2 (x)) = \frac{3}{2} \log (15)$

Cancel the common factor.

$\frac{1}{2} (2 x) = \frac{3}{2} \log (15)$

Rewrite the expression.

$x = \frac{3}{2} \log (15)$

Simplify the right side.

Combine $\frac{3}{2}$ and $\log (15)$ .

$x = \frac{3 \log (15)}{2}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 \log (15)}{2}$

Decimal Form:

$x = 1.76413688 \dots$