Finite Math Examples

$\log (4) + \log (x) = \log (5) - \log (x)$

Step 1

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

$\log (4 x) = \log (5) - \log (x)$

Step 2

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

$\log (4 x) = \log (\frac{5}{x})$

Step 3

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

$4 x = \frac{5}{x}$

Step 4

Solve for $x$ .

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Step 4.1

Find the LCD of the terms in the equation.

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Step 4.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x$

Step 4.1.2

The LCM of one and any expression is the expression.

$x$

Step 4.2

Multiply each term in $4 x = \frac{5}{x}$ by $x$ to eliminate the fractions.

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Step 4.2.1

Multiply each term in $4 x = \frac{5}{x}$ by $x$ .

$4 x \cdot x = \frac{5}{x} x$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

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

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Step 4.2.2.1.1

Move $x$ .

$4 (x \cdot x) = \frac{5}{x} x$

Step 4.2.2.1.2

Multiply $x$ by $x$ .

$4 x^{2} = \frac{5}{x} x$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Cancel the common factor of $x$ .

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Step 4.2.3.1.1

Cancel the common factor.

$4 x^{2} = \frac{5}{x} x$

Step 4.2.3.1.2

Rewrite the expression.

$4 x^{2} = 5$

Step 4.3

Solve the equation.

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Step 4.3.1

Divide each term in $4 x^{2} = 5$ by $4$ and simplify.

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Step 4.3.1.1

Divide each term in $4 x^{2} = 5$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{5}{4}$

Step 4.3.1.2

Simplify the left side.

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Step 4.3.1.2.1

Cancel the common factor of $4$ .

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Step 4.3.1.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{5}{4}$

Step 4.3.1.2.1.2

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

$x^{2} = \frac{5}{4}$

Step 4.3.2

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

$x = \pm \sqrt{\frac{5}{4}}$

Step 4.3.3

Simplify $\pm \sqrt{\frac{5}{4}}$ .

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Step 4.3.3.1

Rewrite $\sqrt{\frac{5}{4}}$ as $\frac{\sqrt{5}}{\sqrt{4}}$ .

$x = \pm \frac{\sqrt{5}}{\sqrt{4}}$

Step 4.3.3.2

Simplify the denominator.

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Step 4.3.3.2.1

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

$x = \pm \frac{\sqrt{5}}{\sqrt{2^{2}}}$

Step 4.3.3.2.2

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

$x = \pm \frac{\sqrt{5}}{2}$

Step 4.3.4

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

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Step 4.3.4.1

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

$x = \frac{\sqrt{5}}{2}$

Step 4.3.4.2

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

$x = - \frac{\sqrt{5}}{2}$

Step 4.3.4.3

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

$x = \frac{\sqrt{5}}{2}, - \frac{\sqrt{5}}{2}$

Step 5

Exclude the solutions that do not make $\log (4) + \log (x) = \log (5) - \log (x)$ true.

$x = \frac{\sqrt{5}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{5}}{2}$

Decimal Form:

$x = 1.11803398 \dots$