Precalculus Examples

$4 \log_{3} (2 x) + 8 \log_{3} (x) - 5 = 0$

Step 1

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

$4 \log_{3} (2 x) + 8 \log_{3} (x) = 5 + 0$

Step 2

Add $5$ and $0$ .

$4 \log_{3} (2 x) + 8 \log_{3} (x) = 5$

Step 3

Simplify the left side.

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Step 3.1

Simplify $4 \log_{3} (2 x) + 8 \log_{3} (x)$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Simplify $4 \log_{3} (2 x)$ by moving $4$ inside the logarithm.

$\log_{3} ({(2 x)}^{4}) + 8 \log_{3} (x) = 5$

Step 3.1.1.2

Apply the product rule to $2 x$ .

$\log_{3} (2^{4} x^{4}) + 8 \log_{3} (x) = 5$

Step 3.1.1.3

Raise $2$ to the power of $4$ .

$\log_{3} (16 x^{4}) + 8 \log_{3} (x) = 5$

Step 3.1.1.4

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

$\log_{3} (16 x^{4}) + \log_{3} (x^{8}) = 5$

Step 3.1.2

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

$\log_{3} (16 x^{4} x^{8}) = 5$

Step 3.1.3

Multiply $x^{4}$ by $x^{8}$ by adding the exponents.

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Step 3.1.3.1

Move $x^{8}$ .

$\log_{3} (16 (x^{8} x^{4})) = 5$

Step 3.1.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{3} (16 x^{8 + 4}) = 5$

Step 3.1.3.3

Add $8$ and $4$ .

$\log_{3} (16 x^{12}) = 5$

Step 4

Rewrite $\log_{3} (16 x^{12}) = 5$ 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 $b^{y} = x$ .

$3^{5} = 16 x^{12}$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $16 x^{12} = 3^{5}$ .

$16 x^{12} = 3^{5}$

Step 5.2

Divide each term in $16 x^{12} = 3^{5}$ by $16$ and simplify.

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Step 5.2.1

Divide each term in $16 x^{12} = 3^{5}$ by $16$ .

$\frac{16 x^{12}}{16} = \frac{3^{5}}{16}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $16$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{16 x^{12}}{16} = \frac{3^{5}}{16}$

Step 5.2.2.1.2

Divide $x^{12}$ by $1$ .

$x^{12} = \frac{3^{5}}{16}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Raise $3$ to the power of $5$ .

$x^{12} = \frac{243}{16}$

Step 5.3

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

$x = \pm \sqrt[12]{\frac{243}{16}}$

Step 5.4

Simplify $\pm \sqrt[12]{\frac{243}{16}}$ .

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Step 5.4.1

Rewrite $\sqrt[12]{\frac{243}{16}}$ as $\frac{\sqrt[12]{243}}{\sqrt[12]{16}}$ .

$x = \pm \frac{\sqrt[12]{243}}{\sqrt[12]{16}}$

Step 5.4.2

Simplify the denominator.

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Step 5.4.2.1

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

$x = \pm \frac{\sqrt[12]{243}}{\sqrt[12]{2^{4}}}$

Step 5.4.2.2

Rewrite $\sqrt[12]{2^{4}}$ as $\sqrt[3]{\sqrt[4]{2^{4}}}$ .

$x = \pm \frac{\sqrt[12]{243}}{\sqrt[3]{\sqrt[4]{2^{4}}}}$

Step 5.4.2.3

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

$x = \pm \frac{\sqrt[12]{243}}{\sqrt[3]{2}}$

Step 5.4.3

Multiply $\frac{\sqrt[12]{243}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$x = \pm \frac{\sqrt[12]{243}}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$

Step 5.4.4

Combine and simplify the denominator.

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Step 5.4.4.1

Multiply $\frac{\sqrt[12]{243}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$x = \pm \frac{\sqrt[12]{243} {\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}}$

Step 5.4.4.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt[12]{243} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1} {\sqrt[3]{2}}^{2}}$

Step 5.4.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt[12]{243} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}}$

Step 5.4.4.4

Add $1$ and $2$ .

$x = \pm \frac{\sqrt[12]{243} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{3}}$

Step 5.4.4.5

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

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Step 5.4.4.5.1

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

$x = \pm \frac{\sqrt[12]{243} {\sqrt[3]{2}}^{2}}{{(2^{\frac{1}{3}})}^{3}}$

Step 5.4.4.5.2

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

$x = \pm \frac{\sqrt[12]{243} {\sqrt[3]{2}}^{2}}{2^{\frac{1}{3} \cdot 3}}$

Step 5.4.4.5.3

Combine $\frac{1}{3}$ and $3$ .

$x = \pm \frac{\sqrt[12]{243} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 5.4.4.5.4

Cancel the common factor of $3$ .

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Step 5.4.4.5.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt[12]{243} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 5.4.4.5.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt[12]{243} {\sqrt[3]{2}}^{2}}{2^{1}}$

Step 5.4.4.5.5

Evaluate the exponent.

$x = \pm \frac{\sqrt[12]{243} {\sqrt[3]{2}}^{2}}{2}$

Step 5.4.5

Simplify the numerator.

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Step 5.4.5.1

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

$x = \pm \frac{\sqrt[12]{243} \sqrt[3]{2^{2}}}{2}$

Step 5.4.5.2

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

$x = \pm \frac{\sqrt[12]{243} \sqrt[3]{4}}{2}$

Step 5.4.6

Simplify the numerator.

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Step 5.4.6.1

Rewrite the expression using the least common index of $12$ .

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Step 5.4.6.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$x = \pm \frac{\sqrt[12]{243} \cdot 4^{\frac{1}{3}}}{2}$

Step 5.4.6.1.2

Rewrite $4^{\frac{1}{3}}$ as $4^{\frac{4}{12}}$ .

$x = \pm \frac{\sqrt[12]{243} \cdot 4^{\frac{4}{12}}}{2}$

Step 5.4.6.1.3

Rewrite $4^{\frac{4}{12}}$ as $\sqrt[12]{4^{4}}$ .

$x = \pm \frac{\sqrt[12]{243} \sqrt[12]{4^{4}}}{2}$

Step 5.4.6.2

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt[12]{243 \cdot 4^{4}}}{2}$

Step 5.4.6.3

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

$x = \pm \frac{\sqrt[12]{243 \cdot 256}}{2}$

Step 5.4.7

Multiply $243$ by $256$ .

$x = \pm \frac{\sqrt[12]{62208}}{2}$

Step 5.5

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

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Step 5.5.1

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

$x = \frac{\sqrt[12]{62208}}{2}$

Step 5.5.2

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

$x = - \frac{\sqrt[12]{62208}}{2}$

Step 5.5.3

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

$x = \frac{\sqrt[12]{62208}}{2}, - \frac{\sqrt[12]{62208}}{2}$

Step 6

Exclude the solutions that do not make $4 \log_{3} (2 x) + 8 \log_{3} (x) - 5 = 0$ true.

$x = \frac{\sqrt[12]{62208}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt[12]{62208}}{2}$

Decimal Form:

$x = 1.25446107 \dots$