Precalculus Examples

$\log (x) + \log (2 x) = 5$

Step 1

Simplify the left side.

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Step 1.1

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

$\log (x (2 x)) = 5$

Step 1.2

Rewrite using the commutative property of multiplication.

$\log (2 x \cdot x) = 5$

Step 1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.1

Move $x$ .

$\log (2 (x \cdot x)) = 5$

Step 1.3.2

Multiply $x$ by $x$ .

$\log (2 x^{2}) = 5$

Step 2

Rewrite $\log (2 x^{2}) = 5$ 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 $b^{y} = x$ .

$10^{5} = 2 x^{2}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $2 x^{2} = 10^{5}$ .

$2 x^{2} = 10^{5}$

Step 3.2

Divide each term in $2 x^{2} = 10^{5}$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 x^{2} = 10^{5}$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{10^{5}}{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{10^{5}}{2}$

Step 3.2.2.1.2

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

$x^{2} = \frac{10^{5}}{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Raise $10$ to the power of $5$ .

$x^{2} = \frac{100000}{2}$

Step 3.2.3.2

Divide $100000$ by $2$ .

$x^{2} = 50000$

Step 3.3

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

$x = \pm \sqrt{50000}$

Step 3.4

Simplify $\pm \sqrt{50000}$ .

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Step 3.4.1

Rewrite $50000$ as $100^{2} \cdot 5$ .

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Step 3.4.1.1

Factor $10000$ out of $50000$ .

$x = \pm \sqrt{10000 (5)}$

Step 3.4.1.2

Rewrite $10000$ as $100^{2}$ .

$x = \pm \sqrt{100^{2} \cdot 5}$

Step 3.4.2

Pull terms out from under the radical.

$x = \pm 100 \sqrt{5}$

Step 3.5

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

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Step 3.5.1

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

$x = 100 \sqrt{5}$

Step 3.5.2

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

$x = - 100 \sqrt{5}$

Step 3.5.3

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

$x = 100 \sqrt{5}, - 100 \sqrt{5}$

Step 4

Exclude the solutions that do not make $\log (x) + \log (2 x) = 5$ true.

$x = 100 \sqrt{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 100 \sqrt{5}$

Decimal Form:

$x = 223.60679774 \dots$