Algebra Examples

$\log_{3} (5) + 2 \log_{3} (x) = \log_{3} (125)$

Step 1

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

$\log_{3} (5) + 2 \log_{3} (x) - \log_{3} (125) = 0$

Step 2

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

$2 \log_{3} (x) + \log_{3} (\frac{5}{125}) = 0$

Step 3

Cancel the common factor of $5$ and $125$ .

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Step 3.1

Factor $5$ out of $5$ .

$2 \log_{3} (x) + \log_{3} (\frac{5 (1)}{125}) = 0$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $5$ out of $125$ .

$2 \log_{3} (x) + \log_{3} (\frac{5 \cdot 1}{5 \cdot 25}) = 0$

Step 3.2.2

Cancel the common factor.

$2 \log_{3} (x) + \log_{3} (\frac{5 \cdot 1}{5 \cdot 25}) = 0$

Step 3.2.3

Rewrite the expression.

$2 \log_{3} (x) + \log_{3} (\frac{1}{25}) = 0$

Step 4

Simplify the left side.

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Step 4.1

Simplify $2 \log_{3} (x) + \log_{3} (\frac{1}{25})$ .

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Step 4.1.1

Simplify $2 \log_{3} (x)$ by moving $2$ inside the logarithm.

$\log_{3} (x^{2}) + \log_{3} (\frac{1}{25}) = 0$

Step 4.1.2

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

$\log_{3} (x^{2} \frac{1}{25}) = 0$

Step 4.1.3

Combine $x^{2}$ and $\frac{1}{25}$ .

$\log_{3} (\frac{x^{2}}{25}) = 0$

Step 5

Rewrite $\log_{3} (\frac{x^{2}}{25}) = 0$ 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 $b^{y} = x$ .

$3^{0} = \frac{x^{2}}{25}$

Step 6

Solve for $x$ .

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Step 6.1

Rewrite the equation as $\frac{x^{2}}{25} = 3^{0}$ .

$\frac{x^{2}}{25} = 3^{0}$

Step 6.2

Multiply both sides of the equation by $25$ .

$25 \frac{x^{2}}{25} = 25 \cdot 3^{0}$

Step 6.3

Simplify both sides of the equation.

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Step 6.3.1

Simplify the left side.

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Step 6.3.1.1

Cancel the common factor of $25$ .

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Step 6.3.1.1.1

Cancel the common factor.

$25 \frac{x^{2}}{25} = 25 \cdot 3^{0}$

Step 6.3.1.1.2

Rewrite the expression.

$x^{2} = 25 \cdot 3^{0}$

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

Simplify $25 \cdot 3^{0}$ .

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Step 6.3.2.1.1

Anything raised to $0$ is $1$ .

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

Step 6.3.2.1.2

Multiply $25$ by $1$ .

$x^{2} = 25$

Step 6.4

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

$x = \pm \sqrt{25}$

Step 6.5

Simplify $\pm \sqrt{25}$ .

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Step 6.5.1

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

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

Step 6.5.2

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

$x = \pm 5$

Step 6.6

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

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Step 6.6.1

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

$x = 5$

Step 6.6.2

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

$x = - 5$

Step 6.6.3

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

$x = 5, - 5$

Step 7

Exclude the solutions that do not make $\log_{3} (5) + 2 \log_{3} (x) = \log_{3} (125)$ true.

$x = 5$