Algebra Examples

$2 \log (4) - \log (3) + 2 \log (x - 4) = 0$

Step 1

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

$2 \log (4) - \log (3) + 2 \log (x - 4) = 0$

Step 2

Simplify the left side.

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Step 2.1

Simplify $2 \log (4) - \log (3) + 2 \log (x - 4)$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

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

$\log (4^{2}) - \log (3) + 2 \log (x - 4) = 0$

Step 2.1.1.2

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

$\log (16) - \log (3) + 2 \log (x - 4) = 0$

Step 2.1.1.3

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

$\log (16) - \log (3) + \log ({(x - 4)}^{2}) = 0$

Step 2.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{16}{3}) + \log ({(x - 4)}^{2}) = 0$

Step 2.1.3

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

$\log (\frac{16}{3} {(x - 4)}^{2}) = 0$

Step 2.1.4

Combine $\frac{16}{3}$ and ${(x - 4)}^{2}$ .

$\log (\frac{16 {(x - 4)}^{2}}{3}) = 0$

Step 3

Rewrite $\log (\frac{16 {(x - 4)}^{2}}{3}) = 0$ 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 $b^{y} = x$ .

$10^{0} = \frac{16 {(x - 4)}^{2}}{3}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $\frac{16 {(x - 4)}^{2}}{3} = 10^{0}$ .

$\frac{16 {(x - 4)}^{2}}{3} = 10^{0}$

Step 4.2

Multiply both sides of the equation by $\frac{3}{16}$ .

$\frac{3}{16} \cdot \frac{16 {(x - 4)}^{2}}{3} = \frac{3}{16} \cdot 10^{0}$

Step 4.3

Simplify both sides of the equation.

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Step 4.3.1

Simplify the left side.

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Step 4.3.1.1

Simplify $\frac{3}{16} \cdot \frac{16 {(x - 4)}^{2}}{3}$ .

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Step 4.3.1.1.1

Combine.

$\frac{3 (16 {(x - 4)}^{2})}{16 \cdot 3} = \frac{3}{16} \cdot 10^{0}$

Step 4.3.1.1.2

Cancel the common factor of $3$ .

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Step 4.3.1.1.2.1

Cancel the common factor.

$\frac{3 (16 {(x - 4)}^{2})}{16 \cdot 3} = \frac{3}{16} \cdot 10^{0}$

Step 4.3.1.1.2.2

Rewrite the expression.

$\frac{16 {(x - 4)}^{2}}{16} = \frac{3}{16} \cdot 10^{0}$

Step 4.3.1.1.3

Cancel the common factor of $16$ .

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Step 4.3.1.1.3.1

Cancel the common factor.

$\frac{16 {(x - 4)}^{2}}{16} = \frac{3}{16} \cdot 10^{0}$

Step 4.3.1.1.3.2

Divide ${(x - 4)}^{2}$ by $1$ .

${(x - 4)}^{2} = \frac{3}{16} \cdot 10^{0}$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Simplify $\frac{3}{16} \cdot 10^{0}$ .

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Step 4.3.2.1.1

Anything raised to $0$ is $1$ .

${(x - 4)}^{2} = \frac{3}{16} \cdot 1$

Step 4.3.2.1.2

Multiply $\frac{3}{16}$ by $1$ .

${(x - 4)}^{2} = \frac{3}{16}$

Step 4.4

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

$x - 4 = \pm \sqrt{\frac{3}{16}}$

Step 4.5

Simplify $\pm \sqrt{\frac{3}{16}}$ .

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Step 4.5.1

Rewrite $\sqrt{\frac{3}{16}}$ as $\frac{\sqrt{3}}{\sqrt{16}}$ .

$x - 4 = \pm \frac{\sqrt{3}}{\sqrt{16}}$

Step 4.5.2

Simplify the denominator.

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Step 4.5.2.1

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

$x - 4 = \pm \frac{\sqrt{3}}{\sqrt{4^{2}}}$

Step 4.5.2.2

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

$x - 4 = \pm \frac{\sqrt{3}}{4}$

Step 4.6

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

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Step 4.6.1

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

$x - 4 = \frac{\sqrt{3}}{4}$

Step 4.6.2

Add $4$ to both sides of the equation.

$x = \frac{\sqrt{3}}{4} + 4$

Step 4.6.3

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

$x - 4 = - \frac{\sqrt{3}}{4}$

Step 4.6.4

Add $4$ to both sides of the equation.

$x = - \frac{\sqrt{3}}{4} + 4$

Step 4.6.5

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

$x = \frac{\sqrt{3}}{4} + 4, - \frac{\sqrt{3}}{4} + 4$

Step 5

Exclude the solutions that do not make $2 \log (4) - \log (3) + 2 \log (x - 4) = 0$ true.

$x = \frac{\sqrt{3}}{4} + 4$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{3}}{4} + 4$

Decimal Form:

$x = 4.43301270 \dots$