Algebra Examples

$\log_{5} (\frac{25}{x}) = 4 + \log_{5} (x)$

Step 1

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

$\log_{5} (\frac{25}{x}) - \log_{5} (x) = 4$

Step 2

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

$\log_{5} (\frac{\frac{25}{x}}{x}) = 4$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\log_{5} (\frac{25}{x} \cdot \frac{1}{x}) = 4$

Step 4

Multiply $\frac{25}{x} \cdot \frac{1}{x}$ .

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Step 4.1

Multiply $\frac{25}{x}$ by $\frac{1}{x}$ .

$\log_{5} (\frac{25}{x \cdot x}) = 4$

Step 4.2

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

$\log_{5} (\frac{25}{x \cdot x}) = 4$

Step 4.3

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

$\log_{5} (\frac{25}{x \cdot x}) = 4$

Step 4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{5} (\frac{25}{x^{1 + 1}}) = 4$

Step 4.5

Add $1$ and $1$ .

$\log_{5} (\frac{25}{x^{2}}) = 4$

Step 5

Rewrite $\log_{5} (\frac{25}{x^{2}}) = 4$ 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 $b^{y} = x$ .

$5^{4} = \frac{25}{x^{2}}$

Step 6

Solve for $x$ .

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Step 6.1

Rewrite the equation as $\frac{25}{x^{2}} = 5^{4}$ .

$\frac{25}{x^{2}} = 5^{4}$

Step 6.2

Raise $5$ to the power of $4$ .

$\frac{25}{x^{2}} = 625$

Step 6.3

Find the LCD of the terms in the equation.

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Step 6.3.1

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

$x^{2}, 1$

Step 6.3.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 6.4

Multiply each term in $\frac{25}{x^{2}} = 625$ by $x^{2}$ to eliminate the fractions.

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Step 6.4.1

Multiply each term in $\frac{25}{x^{2}} = 625$ by $x^{2}$ .

$\frac{25}{x^{2}} x^{2} = 625 x^{2}$

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Cancel the common factor of $x^{2}$ .

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Step 6.4.2.1.1

Cancel the common factor.

$\frac{25}{x^{2}} x^{2} = 625 x^{2}$

Step 6.4.2.1.2

Rewrite the expression.

$25 = 625 x^{2}$

Step 6.5

Solve the equation.

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Step 6.5.1

Rewrite the equation as $625 x^{2} = 25$ .

$625 x^{2} = 25$

Step 6.5.2

Divide each term in $625 x^{2} = 25$ by $625$ and simplify.

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Step 6.5.2.1

Divide each term in $625 x^{2} = 25$ by $625$ .

$\frac{625 x^{2}}{625} = \frac{25}{625}$

Step 6.5.2.2

Simplify the left side.

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Step 6.5.2.2.1

Cancel the common factor of $625$ .

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Step 6.5.2.2.1.1

Cancel the common factor.

$\frac{625 x^{2}}{625} = \frac{25}{625}$

Step 6.5.2.2.1.2

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

$x^{2} = \frac{25}{625}$

Step 6.5.2.3

Simplify the right side.

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Step 6.5.2.3.1

Cancel the common factor of $25$ and $625$ .

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Step 6.5.2.3.1.1

Factor $25$ out of $25$ .

$x^{2} = \frac{25 (1)}{625}$

Step 6.5.2.3.1.2

Cancel the common factors.

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Step 6.5.2.3.1.2.1

Factor $25$ out of $625$ .

$x^{2} = \frac{25 \cdot 1}{25 \cdot 25}$

Step 6.5.2.3.1.2.2

Cancel the common factor.

$x^{2} = \frac{25 \cdot 1}{25 \cdot 25}$

Step 6.5.2.3.1.2.3

Rewrite the expression.

$x^{2} = \frac{1}{25}$

Step 6.5.3

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

$x = \pm \sqrt{\frac{1}{25}}$

Step 6.5.4

Simplify $\pm \sqrt{\frac{1}{25}}$ .

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Step 6.5.4.1

Rewrite $\sqrt{\frac{1}{25}}$ as $\frac{\sqrt{1}}{\sqrt{25}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{25}}$

Step 6.5.4.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{25}}$

Step 6.5.4.3

Simplify the denominator.

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Step 6.5.4.3.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \frac{1}{\sqrt{5^{2}}}$

Step 6.5.4.3.2

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

$x = \pm \frac{1}{5}$

Step 6.5.5

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

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Step 6.5.5.1

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

$x = \frac{1}{5}$

Step 6.5.5.2

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

$x = - \frac{1}{5}$

Step 6.5.5.3

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

$x = \frac{1}{5}, - \frac{1}{5}$

Step 7

Exclude the solutions that do not make $\log_{5} (\frac{25}{x}) = 4 + \log_{5} (x)$ true.

$x = \frac{1}{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{5}$

Decimal Form:

$x = 0.2$