Algebra Examples

$\log_{7} (3 x^{3} + x) - \log_{7} (x) = 2$

Step 1

Simplify the left side.

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Step 1.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{7} (\frac{3 x^{3} + x}{x}) = 2$

Step 1.2

Factor $x$ out of $3 x^{3} + x$ .

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Step 1.2.1

Factor $x$ out of $3 x^{3}$ .

$\log_{7} (\frac{x (3 x^{2}) + x}{x}) = 2$

Step 1.2.2

Raise $x$ to the power of $1$ .

$\log_{7} (\frac{x (3 x^{2}) + x}{x}) = 2$

Step 1.2.3

Factor $x$ out of $x^{1}$ .

$\log_{7} (\frac{x (3 x^{2}) + x \cdot 1}{x}) = 2$

Step 1.2.4

Factor $x$ out of $x (3 x^{2}) + x \cdot 1$ .

$\log_{7} (\frac{x (3 x^{2} + 1)}{x}) = 2$

Step 1.3

Cancel the common factor of $x$ .

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Step 1.3.1

Cancel the common factor.

$\log_{7} (\frac{x (3 x^{2} + 1)}{x}) = 2$

Step 1.3.2

Divide $3 x^{2} + 1$ by $1$ .

$\log_{7} (3 x^{2} + 1) = 2$

Step 2

Rewrite $\log_{7} (3 x^{2} + 1) = 2$ 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 $b^{y} = x$ .

$7^{2} = 3 x^{2} + 1$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $3 x^{2} + 1 = 7^{2}$ .

$3 x^{2} + 1 = 7^{2}$

Step 3.2

Raise $7$ to the power of $2$ .

$3 x^{2} + 1 = 49$

Step 3.3

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

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Step 3.3.1

Subtract $1$ from both sides of the equation.

$3 x^{2} = 49 - 1$

Step 3.3.2

Subtract $1$ from $49$ .

$3 x^{2} = 48$

Step 3.4

Divide each term in $3 x^{2} = 48$ by $3$ and simplify.

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Step 3.4.1

Divide each term in $3 x^{2} = 48$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{48}{3}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $3$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{48}{3}$

Step 3.4.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{48}{3}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $48$ by $3$ .

$x^{2} = 16$

Step 3.5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{16}$

Step 3.6

Simplify $\pm \sqrt{16}$ .

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Step 3.6.1

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt{4^{2}}$

Step 3.6.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 4$

Step 3.7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.7.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 4$

Step 3.7.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 4$

Step 3.7.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 4, - 4$

Step 4

Exclude the solutions that do not make $\log_{7} (3 x^{3} + x) - \log_{7} (x) = 2$ true.

$x = 4$