Algebra Examples

$2 \log (x) - \log (4) = 2$

Step 1

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

$2 \log (x) - \log (4) = 2$

Step 2

Simplify the left side.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 3

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

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

Step 4

Solve for $x$ .

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Step 4.1

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

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

Step 4.2

Multiply both sides of the equation by $4$ .

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

Step 4.3

Simplify the left side.

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Step 4.3.1

Cancel the common factor of $4$ .

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Step 4.3.1.1

Cancel the common factor.

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

Step 4.3.1.2

Rewrite the expression.

$x^{2} = 4 \cdot 10^{2}$

Step 4.4

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

$x = \pm \sqrt{4 \cdot 10^{2}}$

Step 4.5

Simplify $\pm \sqrt{4 \cdot 10^{2}}$ .

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Step 4.5.1

Rewrite $\sqrt{4 \cdot 10^{2}}$ as $\sqrt{4} \cdot \sqrt{10^{2}}$ .

$x = \pm \sqrt{4} \cdot \sqrt{10^{2}}$

Step 4.5.2

Evaluate the root.

$x = \pm 2 \cdot \sqrt{10^{2}}$

Step 4.5.3

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

$x = \pm 2 \cdot 10$

Step 4.5.4

Raise $10$ to the power of $1$ .

$x = \pm 2 \cdot 10^{1}$

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.

$x = 2 \cdot 10^{1}$

Step 4.6.2

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

$x = - 2 \cdot 10^{1}$

Step 4.6.3

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

$x = 2 \cdot 10^{1}, - 2 \cdot 10^{1}$

Step 5

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

$x = 2 \cdot 10^{1}$

Step 6

The result can be shown in multiple forms.

Scientific Notation:

$x = 2 \cdot 10^{1}$

Expanded Form:

$x = 20$