Precalculus Examples

$2 \log_{8} (x - 2) + \log_{8} (16) = 2$

Step 1

Simplify the left side.

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Step 1.1

Logarithm base $8$ of $16$ is $\frac{4}{3}$ .

$2 \log_{8} (x - 2) + \frac{4}{3} = 2$

Step 1.2

To write $2 \log_{8} (x - 2)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 \log_{8} (x - 2) \cdot \frac{3}{3} + \frac{4}{3} = 2$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine $2 \log_{8} (x - 2)$ and $\frac{3}{3}$ .

$\frac{2 \log_{8} (x - 2) \cdot 3}{3} + \frac{4}{3} = 2$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{2 \log_{8} (x - 2) \cdot 3 + 4}{3} = 2$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Factor $2$ out of $2 \log_{8} (x - 2) \cdot 3 + 4$ .

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Step 1.4.1.1

Factor $2$ out of $2 \log_{8} (x - 2) \cdot 3$ .

$\frac{2 (\log_{8} (x - 2) \cdot 3) + 4}{3} = 2$

Step 1.4.1.2

Factor $2$ out of $4$ .

$\frac{2 (\log_{8} (x - 2) \cdot 3) + 2 \cdot 2}{3} = 2$

Step 1.4.1.3

Factor $2$ out of $2 (\log_{8} (x - 2) \cdot 3) + 2 \cdot 2$ .

$\frac{2 (\log_{8} (x - 2) \cdot 3 + 2)}{3} = 2$

Step 1.4.2

Move $3$ to the left of $\log_{8} (x - 2)$ .

$\frac{2 (3 \log_{8} (x - 2) + 2)}{3} = 2$

Step 2

Multiply both sides by $3$ .

$\frac{2 (3 \log_{8} (x - 2) + 2)}{3} \cdot 3 = 2 \cdot 3$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{2 (3 \log_{8} (x - 2) + 2)}{3} \cdot 3$ .

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Step 3.1.1.1

Cancel the common factor of $3$ .

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Step 3.1.1.1.1

Simplify $3 \log_{8} (x - 2)$ by moving $3$ inside the logarithm.

$\frac{2 (\log_{8} ({(x - 2)}^{3}) + 2)}{3} \cdot 3 = 2 \cdot 3$

Step 3.1.1.1.2

Cancel the common factor.

$\frac{2 (\log_{8} ({(x - 2)}^{3}) + 2)}{3} \cdot 3 = 2 \cdot 3$

Step 3.1.1.1.3

Rewrite the expression.

$2 (\log_{8} ({(x - 2)}^{3}) + 2) = 2 \cdot 3$

Step 3.1.1.2

Apply the distributive property.

$2 \log_{8} ({(x - 2)}^{3}) + 2 \cdot 2 = 2 \cdot 3$

Step 3.1.1.3

Simplify $2 \log_{8} ({(x - 2)}^{3})$ by moving $2$ inside the logarithm.

$\log_{8} ({({(x - 2)}^{3})}^{2}) + 2 \cdot 2 = 2 \cdot 3$

Step 3.1.1.4

Multiply $2$ by $2$ .

$\log_{8} ({({(x - 2)}^{3})}^{2}) + 4 = 2 \cdot 3$

Step 3.1.1.5

Multiply the exponents in ${({(x - 2)}^{3})}^{2}$ .

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Step 3.1.1.5.1

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

$\log_{8} ({(x - 2)}^{3 \cdot 2}) + 4 = 2 \cdot 3$

Step 3.1.1.5.2

Multiply $3$ by $2$ .

$\log_{8} ({(x - 2)}^{6}) + 4 = 2 \cdot 3$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $2$ by $3$ .

$\log_{8} ({(x - 2)}^{6}) + 4 = 6$

Step 4

Solve for $x$ .

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Step 4.1

Move all terms not containing $\log_{8} ({(x - 2)}^{6})$ to the right side of the equation.

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Step 4.1.1

Subtract $4$ from both sides of the equation.

$\log_{8} ({(x - 2)}^{6}) = 6 - 4$

Step 4.1.2

Subtract $4$ from $6$ .

$\log_{8} ({(x - 2)}^{6}) = 2$

Step 4.2

Rewrite $\log_{8} ({(x - 2)}^{6}) = 2$ 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 $b^{y} = x$ .

$8^{2} = {(x - 2)}^{6}$

Step 4.3

Solve for $x$ .

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Step 4.3.1

Rewrite the equation as ${(x - 2)}^{6} = 8^{2}$ .

${(x - 2)}^{6} = 8^{2}$

Step 4.3.2

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

$x - 2 = \pm \sqrt[6]{8^{2}}$

Step 4.3.3

Simplify $\pm \sqrt[6]{8^{2}}$ .

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Step 4.3.3.1

Raise $8$ to the power of $2$ .

$x - 2 = \pm \sqrt[6]{64}$

Step 4.3.3.2

Rewrite $64$ as $2^{6}$ .

$x - 2 = \pm \sqrt[6]{2^{6}}$

Step 4.3.3.3

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

$x - 2 = \pm 2$

Step 4.3.4

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

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Step 4.3.4.1

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

$x - 2 = 2$

Step 4.3.4.2

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

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Step 4.3.4.2.1

Add $2$ to both sides of the equation.

$x = 2 + 2$

Step 4.3.4.2.2

Add $2$ and $2$ .

$x = 4$

Step 4.3.4.3

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

$x - 2 = - 2$

Step 4.3.4.4

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

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Step 4.3.4.4.1

Add $2$ to both sides of the equation.

$x = - 2 + 2$

Step 4.3.4.4.2

Add $- 2$ and $2$ .

$x = 0$

Step 4.3.4.5

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

$x = 4, 0$

Step 5

Exclude the solutions that do not make $2 \log_{8} (x - 2) + \log_{8} (16) = 2$ true.

$x = 4$