Precalculus Examples

$2 \log_{6} (x - 5) + \log_{6} (4) = 2$

Step 1

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

$2 \log_{6} (x - 5) + \log_{6} (4) = 2$

Step 2

Simplify the left side.

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Step 2.1

Simplify $2 \log_{6} (x - 5) + \log_{6} (4)$ .

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Step 2.1.1

Simplify $2 \log_{6} (x - 5)$ by moving $2$ inside the logarithm.

$\log_{6} ({(x - 5)}^{2}) + \log_{6} (4) = 2$

Step 2.1.2

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

$\log_{6} ({(x - 5)}^{2} \cdot 4) = 2$

Step 2.1.3

Move $4$ to the left of ${(x - 5)}^{2}$ .

$\log_{6} (4 {(x - 5)}^{2}) = 2$

Step 3

Rewrite $\log_{6} (4 {(x - 5)}^{2}) = 2$ 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 $b^{y} = x$ .

$6^{2} = 4 {(x - 5)}^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $4 {(x - 5)}^{2} = 6^{2}$ .

$4 {(x - 5)}^{2} = 6^{2}$

Step 4.2

Divide each term in $4 {(x - 5)}^{2} = 6^{2}$ by $4$ and simplify.

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Step 4.2.1

Divide each term in $4 {(x - 5)}^{2} = 6^{2}$ by $4$ .

$\frac{4 {(x - 5)}^{2}}{4} = \frac{6^{2}}{4}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{4 {(x - 5)}^{2}}{4} = \frac{6^{2}}{4}$

Step 4.2.2.1.2

Divide ${(x - 5)}^{2}$ by $1$ .

${(x - 5)}^{2} = \frac{6^{2}}{4}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Raise $6$ to the power of $2$ .

${(x - 5)}^{2} = \frac{36}{4}$

Step 4.2.3.2

Divide $36$ by $4$ .

${(x - 5)}^{2} = 9$

Step 4.3

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

$x - 5 = \pm \sqrt{9}$

Step 4.4

Simplify $\pm \sqrt{9}$ .

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Step 4.4.1

Rewrite $9$ as $3^{2}$ .

$x - 5 = \pm \sqrt{3^{2}}$

Step 4.4.2

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

$x - 5 = \pm 3$

Step 4.5

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

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Step 4.5.1

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

$x - 5 = 3$

Step 4.5.2

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

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Step 4.5.2.1

Add $5$ to both sides of the equation.

$x = 3 + 5$

Step 4.5.2.2

Add $3$ and $5$ .

$x = 8$

Step 4.5.3

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

$x - 5 = - 3$

Step 4.5.4

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

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Step 4.5.4.1

Add $5$ to both sides of the equation.

$x = - 3 + 5$

Step 4.5.4.2

Add $- 3$ and $5$ .

$x = 2$

Step 4.5.5

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

$x = 8, 2$

Step 5

Exclude the solutions that do not make $2 \log_{6} (x - 5) + \log_{6} (4) = 2$ true.

$x = 8$