Algebra Examples

$2 \log (\sqrt{3}) + \log (x - 3) = 3$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$2 \log (\sqrt{3}) + \log (x - 3) = 3$

Step 2

Simplify the left side.

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Step 2.1

Simplify $2 \log (\sqrt{3}) + \log (x - 3)$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Simplify $2 \log (\sqrt{3})$ by moving $2$ inside the logarithm.

$\log ({\sqrt{3}}^{2}) + \log (x - 3) = 3$

Step 2.1.1.2

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

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Step 2.1.1.2.1

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

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

Step 2.1.1.2.2

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

$\log (3^{\frac{1}{2} \cdot 2}) + \log (x - 3) = 3$

Step 2.1.1.2.3

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

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

Step 2.1.1.2.4

Cancel the common factor of $2$ .

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Step 2.1.1.2.4.1

Cancel the common factor.

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

Step 2.1.1.2.4.2

Rewrite the expression.

$\log (3^{1}) + \log (x - 3) = 3$

Step 2.1.1.2.5

Evaluate the exponent.

$\log (3) + \log (x - 3) = 3$

Step 2.1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (3 (x - 3)) = 3$

Step 2.1.3

Apply the distributive property.

$\log (3 x + 3 \cdot - 3) = 3$

Step 2.1.4

Multiply $3$ by $- 3$ .

$\log (3 x - 9) = 3$

Step 3

Rewrite $\log (3 x - 9) = 3$ 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 $b^{y} = x$ .

$10^{3} = 3 x - 9$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $3 x - 9 = 10^{3}$ .

$3 x - 9 = 10^{3}$

Step 4.2

Raise $10$ to the power of $3$ .

$3 x - 9 = 1000$

Step 4.3

Move all terms not containing $x$ to the right side of the equation.

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Step 4.3.1

Add $9$ to both sides of the equation.

$3 x = 1000 + 9$

Step 4.3.2

Add $1000$ and $9$ .

$3 x = 1009$

Step 4.4

Divide each term in $3 x = 1009$ by $3$ and simplify.

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Step 4.4.1

Divide each term in $3 x = 1009$ by $3$ .

$\frac{3 x}{3} = \frac{1009}{3}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $3$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{1009}{3}$

Step 4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{1009}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1009}{3}$

Decimal Form:

$x = 336.\overline{3}$

Mixed Number Form:

$x = 336 \frac{1}{3}$