Algebra Examples

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

Step 1

Simplify.

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Step 1.1

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

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

Step 1.2

Simplify $2 \ln (8)$ by moving $2$ inside the logarithm.

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

Step 2

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

${(4 x)}^{2} = 8^{2}$

Step 3

Solve for $x$ .

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Step 3.1

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| 4 x | = | 8 |$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$4 x = \pm | 8 |$

Step 3.2.2

The absolute value is the distance between a number and zero. The distance between $0$ and $8$ is $8$ .

$4 x = \pm 8$

Step 3.2.3

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

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Step 3.2.3.1

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

$4 x = 8$

Step 3.2.3.2

Divide each term in $4 x = 8$ by $4$ and simplify.

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Step 3.2.3.2.1

Divide each term in $4 x = 8$ by $4$ .

$\frac{4 x}{4} = \frac{8}{4}$

Step 3.2.3.2.2

Simplify the left side.

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Step 3.2.3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.3.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{8}{4}$

Step 3.2.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{4}$

Step 3.2.3.2.3

Simplify the right side.

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Step 3.2.3.2.3.1

Divide $8$ by $4$ .

$x = 2$

Step 3.2.3.3

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

$4 x = - 8$

Step 3.2.3.4

Divide each term in $4 x = - 8$ by $4$ and simplify.

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Step 3.2.3.4.1

Divide each term in $4 x = - 8$ by $4$ .

$\frac{4 x}{4} = \frac{- 8}{4}$

Step 3.2.3.4.2

Simplify the left side.

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Step 3.2.3.4.2.1

Cancel the common factor of $4$ .

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Step 3.2.3.4.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 8}{4}$

Step 3.2.3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 8}{4}$

Step 3.2.3.4.3

Simplify the right side.

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Step 3.2.3.4.3.1

Divide $- 8$ by $4$ .

$x = - 2$

Step 3.2.3.5

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

$x = 2, - 2$

Step 4

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

$x = 2$