Algebra Examples

$\log_{\sqrt{3}} (x) - 8 = 0$

Step 1

Add $8$ to both sides of the equation.

$\log_{\sqrt{3}} (x) = 8$

Step 2

Rewrite $\log_{\sqrt{3}} (x) = 8$ 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 $b^{y} = x$ .

${\sqrt{3}}^{8} = x$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $x = {\sqrt{3}}^{8}$ .

$x = {\sqrt{3}}^{8}$

Step 3.2

Simplify ${\sqrt{3}}^{8}$ .

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Step 3.2.1

Rewrite ${\sqrt{3}}^{8}$ as $3^{4}$ .

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.1.3

Combine $\frac{1}{2}$ and $8$ .

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

Step 3.2.1.4

Cancel the common factor of $8$ and $2$ .

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Step 3.2.1.4.1

Factor $2$ out of $8$ .

$x = 3^{\frac{2 \cdot 4}{2}}$

Step 3.2.1.4.2

Cancel the common factors.

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Step 3.2.1.4.2.1

Factor $2$ out of $2$ .

$x = 3^{\frac{2 \cdot 4}{2 (1)}}$

Step 3.2.1.4.2.2

Cancel the common factor.

$x = 3^{\frac{2 \cdot 4}{2 \cdot 1}}$

Step 3.2.1.4.2.3

Rewrite the expression.

$x = 3^{\frac{4}{1}}$

Step 3.2.1.4.2.4

Divide $4$ by $1$ .

$x = 3^{4}$

Step 3.2.2

Raise $3$ to the power of $4$ .

$x = 81$