Finite Math Examples

$\log_{7} (x) + \log_{7} (x - 2) = \log_{7} (24)$

Step 1

Subtract $\log_{7} (24)$ from both sides of the equation.

$\log_{7} (x) + \log_{7} (x - 2) - \log_{7} (24) = 0$

Step 2

Simplify $\log_{7} (x) + \log_{7} (x - 2) - \log_{7} (24)$ .

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Step 2.1

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

$\log_{7} (x (x - 2)) - \log_{7} (24) = 0$

Step 2.2

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

$\log_{7} (\frac{x (x - 2)}{24}) = 0$

Step 3

Rewrite $\log_{7} (\frac{x (x - 2)}{24}) = 0$ 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 $b^{y} = x$ .

$7^{0} = \frac{x (x - 2)}{24}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $\frac{x (x - 2)}{24} = 7^{0}$ .

$\frac{x (x - 2)}{24} = 7^{0}$

Step 4.2

Multiply both sides of the equation by $24$ .

$24 \frac{x (x - 2)}{24} = 24 \cdot 7^{0}$

Step 4.3

Simplify both sides of the equation.

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Step 4.3.1

Simplify the left side.

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Step 4.3.1.1

Simplify $24 \frac{x (x - 2)}{24}$ .

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Step 4.3.1.1.1

Cancel the common factor of $24$ .

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Step 4.3.1.1.1.1

Cancel the common factor.

$24 \frac{x (x - 2)}{24} = 24 \cdot 7^{0}$

Step 4.3.1.1.1.2

Rewrite the expression.

$x (x - 2) = 24 \cdot 7^{0}$

Step 4.3.1.1.2

Apply the distributive property.

$x \cdot x + x \cdot - 2 = 24 \cdot 7^{0}$

Step 4.3.1.1.3

Simplify the expression.

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Step 4.3.1.1.3.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 2 = 24 \cdot 7^{0}$

Step 4.3.1.1.3.2

Move $- 2$ to the left of $x$ .

$x^{2} - 2 x = 24 \cdot 7^{0}$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Simplify $24 \cdot 7^{0}$ .

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Step 4.3.2.1.1

Anything raised to $0$ is $1$ .

$x^{2} - 2 x = 24 \cdot 1$

Step 4.3.2.1.2

Multiply $24$ by $1$ .

$x^{2} - 2 x = 24$

Step 4.4

Subtract $24$ from both sides of the equation.

$x^{2} - 2 x - 24 = 0$

Step 4.5

Factor $x^{2} - 2 x - 24$ using the AC method.

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Step 4.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 24$ and whose sum is $- 2$ .

$- 6, 4$

Step 4.5.2

Write the factored form using these integers.

$(x - 6) (x + 4) = 0$

Step 4.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 6 = 0$

$x + 4 = 0$

Step 4.7

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 4.7.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 4.7.2

Add $6$ to both sides of the equation.

$x = 6$

Step 4.8

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 4.8.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 4.8.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 4.9

The final solution is all the values that make $(x - 6) (x + 4) = 0$ true.

$x = 6, - 4$