Algebra Examples

$\log_{\sqrt{3}} (x) = 2$

Step 1

Rewrite $\log_{\sqrt{3}} (x) = 2$ 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 $b^{y} = x$ .

${\sqrt{3}}^{2} = x$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $x = {\sqrt{3}}^{2}$ .

$x = {\sqrt{3}}^{2}$

Step 2.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 2.2.2

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

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

Step 2.2.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

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

Step 2.2.4.2

Rewrite the expression.

$x = 3^{1}$

Step 2.2.5

Evaluate the exponent.

$x = 3$