Algebra Examples

$3 \ln (2) + \ln (8) = 2 \ln (4 x)$

Step 1

Rewrite the equation as $2 \ln (4 x) = 3 \ln (2) + \ln (8)$ .

$2 \ln (4 x) = 3 \ln (2) + \ln (8)$

Step 2

Simplify the left side.

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Step 2.1

Simplify $2 \ln (4 x)$ .

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Step 2.1.1

Simplify $2 \ln (4 x)$ by moving $2$ inside the logarithm.

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

Step 2.1.2

Apply the product rule to $4 x$ .

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

Step 2.1.3

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

$\ln (16 x^{2}) = 3 \ln (2) + \ln (8)$

Step 3

Simplify the right side.

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Step 3.1

Simplify $3 \ln (2) + \ln (8)$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Simplify $3 \ln (2)$ by moving $3$ inside the logarithm.

$\ln (16 x^{2}) = \ln (2^{3}) + \ln (8)$

Step 3.1.1.2

Raise $2$ to the power of $3$ .

$\ln (16 x^{2}) = \ln (8) + \ln (8)$

Step 3.1.2

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

$\ln (16 x^{2}) = \ln (8 \cdot 8)$

Step 3.1.3

Multiply $8$ by $8$ .

$\ln (16 x^{2}) = \ln (64)$

Step 4

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$16 x^{2} = 64$

Step 5

Solve for $x$ .

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Step 5.1

Divide each term in $16 x^{2} = 64$ by $16$ and simplify.

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Step 5.1.1

Divide each term in $16 x^{2} = 64$ by $16$ .

$\frac{16 x^{2}}{16} = \frac{64}{16}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $16$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{16 x^{2}}{16} = \frac{64}{16}$

Step 5.1.2.1.2

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

$x^{2} = \frac{64}{16}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Divide $64$ by $16$ .

$x^{2} = 4$

Step 5.2

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

$x = \pm \sqrt{4}$

Step 5.3

Simplify $\pm \sqrt{4}$ .

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Step 5.3.1

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

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

Step 5.3.2

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

$x = \pm 2$

Step 5.4

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

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Step 5.4.1

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

$x = 2$

Step 5.4.2

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

$x = - 2$

Step 5.4.3

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

$x = 2, - 2$

Step 6

Exclude the solutions that do not make $3 \ln (2) + \ln (8) = 2 \ln (4 x)$ true.

$x = 2$