Solve for x log base x of 8=3/2

$\log_{x} (8) = \frac{3}{2}$

Rewrite $\log_{x} (8) = \frac{3}{2}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$x^{\frac{3}{2}} = 8$

Solve for $x$ .

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{3}{2}})}^{\frac{2}{3}} = 8^{\frac{2}{3}}$

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{3}{2}})}^{\frac{2}{3}}$ .

Multiply the exponents in ${(x^{\frac{3}{2}})}^{\frac{2}{3}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{3}{2} \cdot \frac{2}{3}} = 8^{\frac{2}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{3}{2} \cdot \frac{2}{3}} = 8^{\frac{2}{3}}$

Rewrite the expression.

$x^{\frac{1}{2} \cdot 2} = 8^{\frac{2}{3}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 8^{\frac{2}{3}}$

Rewrite the expression.

$x^{1} = 8^{\frac{2}{3}}$

Simplify.

$x = 8^{\frac{2}{3}}$

Simplify the right side.

Simplify $8^{\frac{2}{3}}$ .

Simplify the expression.

Rewrite $8$ as $2^{3}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$x = 2^{2}$

Raise $2$ to the power of $2$ .

$x = 4$

Solve for x log base x of 8=3/2

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