Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine $\frac{16}{3}$ and $x^{2}$ .

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

Step 7

Add $4$ to both sides of the equation.

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

Step 8

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

$10^{4} = \frac{16 x^{2}}{3}$

Step 9

Solve for $x$ .

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Step 9.1

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

$\frac{16 x^{2}}{3} = 10^{4}$

Step 9.2

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

$\frac{3}{16} \cdot \frac{16 x^{2}}{3} = \frac{3}{16} \cdot 10^{4}$

Step 9.3

Simplify both sides of the equation.

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Step 9.3.1

Simplify the left side.

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Step 9.3.1.1

Simplify $\frac{3}{16} \cdot \frac{16 x^{2}}{3}$ .

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Step 9.3.1.1.1

Combine.

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

Step 9.3.1.1.2

Cancel the common factor of $3$ .

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Step 9.3.1.1.2.1

Cancel the common factor.

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

Step 9.3.1.1.2.2

Rewrite the expression.

$\frac{16 x^{2}}{16} = \frac{3}{16} \cdot 10^{4}$

Step 9.3.1.1.3

Cancel the common factor of $16$ .

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Step 9.3.1.1.3.1

Cancel the common factor.

$\frac{16 x^{2}}{16} = \frac{3}{16} \cdot 10^{4}$

Step 9.3.1.1.3.2

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

$x^{2} = \frac{3}{16} \cdot 10^{4}$

Step 9.3.2

Simplify the right side.

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Step 9.3.2.1

Simplify $\frac{3}{16} \cdot 10^{4}$ .

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Step 9.3.2.1.1

Raise $10$ to the power of $4$ .

$x^{2} = \frac{3}{16} \cdot 10000$

Step 9.3.2.1.2

Cancel the common factor of $16$ .

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Step 9.3.2.1.2.1

Factor $16$ out of $10000$ .

$x^{2} = \frac{3}{16} \cdot (16 (625))$

Step 9.3.2.1.2.2

Cancel the common factor.

$x^{2} = \frac{3}{16} \cdot (16 \cdot 625)$

Step 9.3.2.1.2.3

Rewrite the expression.

$x^{2} = 3 \cdot 625$

Step 9.3.2.1.3

Multiply $3$ by $625$ .

$x^{2} = 1875$

Step 9.4

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

$x = \pm \sqrt{1875}$

Step 9.5

Simplify $\pm \sqrt{1875}$ .

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Step 9.5.1

Rewrite $1875$ as $25^{2} \cdot 3$ .

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Step 9.5.1.1

Factor $625$ out of $1875$ .

$x = \pm \sqrt{625 (3)}$

Step 9.5.1.2

Rewrite $625$ as $25^{2}$ .

$x = \pm \sqrt{25^{2} \cdot 3}$

Step 9.5.2

Pull terms out from under the radical.

$x = \pm 25 \sqrt{3}$

Step 9.6

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

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Step 9.6.1

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

$x = 25 \sqrt{3}$

Step 9.6.2

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

$x = - 25 \sqrt{3}$

Step 9.6.3

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

$x = 25 \sqrt{3}, - 25 \sqrt{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = 25 \sqrt{3}, - 25 \sqrt{3}$

Decimal Form:

$x = 43.30127018 \dots, - 43.30127018 \dots$