Precalculus Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Multiply $\frac{x}{2 x + 3}$ by $2 x - 3$ .

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

Step 2

Rewrite $\log_{5} (\frac{x (2 x - 3)}{2 x + 3}) = 0$ 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 $b^{y} = x$ .

$5^{0} = \frac{x (2 x - 3)}{2 x + 3}$

Step 3

Cross multiply to remove the fraction.

$x (2 x - 3) = 5^{0} (2 x + 3)$

Step 4

Simplify $5^{0} (2 x + 3)$ .

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Step 4.1

Anything raised to $0$ is $1$ .

$x (2 x - 3) = 1 (2 x + 3)$

Step 4.2

Multiply $2 x + 3$ by $1$ .

$x (2 x - 3) = 2 x + 3$

Step 5

Move all terms containing $x$ to the left side of the equation.

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Step 5.1

Subtract $2 x$ from both sides of the equation.

$x (2 x - 3) - 2 x = 3$

Step 5.2

Simplify each term.

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Step 5.2.1

Apply the distributive property.

$x (2 x) + x \cdot - 3 - 2 x = 3$

Step 5.2.2

Rewrite using the commutative property of multiplication.

$2 x \cdot x + x \cdot - 3 - 2 x = 3$

Step 5.2.3

Move $- 3$ to the left of $x$ .

$2 x \cdot x - 3 \cdot x - 2 x = 3$

Step 5.2.4

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

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Step 5.2.4.1

Move $x$ .

$2 (x \cdot x) - 3 \cdot x - 2 x = 3$

Step 5.2.4.2

Multiply $x$ by $x$ .

$2 x^{2} - 3 \cdot x - 2 x = 3$

$2 x^{2} - 3 x - 2 x = 3$

Step 5.3

Subtract $2 x$ from $- 3 x$ .

$2 x^{2} - 5 x = 3$

Step 6

Factor $x$ out of $2 x^{2} - 5 x$ .

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Step 6.1

Factor $x$ out of $2 x^{2}$ .

$x (2 x) - 5 x = 3$

Step 6.2

Factor $x$ out of $- 5 x$ .

$x (2 x) + x \cdot - 5 = 3$

Step 6.3

Factor $x$ out of $x (2 x) + x \cdot - 5$ .

$x (2 x - 5) = 3$

Step 7

Simplify $x (2 x - 5)$ .

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Step 7.1

Simplify by multiplying through.

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Step 7.1.1

Apply the distributive property.

$x (2 x) + x \cdot - 5 = 3$

Step 7.1.2

Reorder.

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Step 7.1.2.1

Rewrite using the commutative property of multiplication.

$2 x \cdot x + x \cdot - 5 = 3$

Step 7.1.2.2

Move $- 5$ to the left of $x$ .

$2 x \cdot x - 5 \cdot x = 3$

Step 7.2

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

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Step 7.2.1

Move $x$ .

$2 (x \cdot x) - 5 \cdot x = 3$

Step 7.2.2

Multiply $x$ by $x$ .

$2 x^{2} - 5 \cdot x = 3$

$2 x^{2} - 5 x = 3$

Step 8

Subtract $3$ from both sides of the equation.

$2 x^{2} - 5 x - 3 = 0$

Step 9

Factor by grouping.

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Step 9.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = - 5$ .

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Step 9.1.1

Factor $- 5$ out of $- 5 x$ .

$2 x^{2} - 5 x - 3 = 0$

Step 9.1.2

Rewrite $- 5$ as $1$ plus $- 6$

$2 x^{2} + (1 - 6) x - 3 = 0$

Step 9.1.3

Apply the distributive property.

$2 x^{2} + 1 x - 6 x - 3 = 0$

Step 9.1.4

Multiply $x$ by $1$ .

$2 x^{2} + x - 6 x - 3 = 0$

Step 9.2

Factor out the greatest common factor from each group.

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Step 9.2.1

Group the first two terms and the last two terms.

$(2 x^{2} + x) - 6 x - 3 = 0$

Step 9.2.2

Factor out the greatest common factor (GCF) from each group.

$x (2 x + 1) - 3 (2 x + 1) = 0$

Step 9.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$(2 x + 1) (x - 3) = 0$

Step 10

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

$2 x + 1 = 0$

$x - 3 = 0$

Step 11

Set $2 x + 1$ equal to $0$ and solve for $x$ .

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Step 11.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 11.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 11.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 11.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 11.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 11.2.2.2

Simplify the left side.

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Step 11.2.2.2.1

Cancel the common factor of $2$ .

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Step 11.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 11.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Step 11.2.2.3

Simplify the right side.

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Step 11.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 12

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 12.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 12.2

Add $3$ to both sides of the equation.

$x = 3$

Step 13

The final solution is all the values that make $(2 x + 1) (x - 3) = 0$ true.

$x = - \frac{1}{2}, 3$

Step 14

Exclude the solutions that do not make $\log_{5} (x) - \log_{5} (2 x + 3) + \log_{5} (2 x - 3) = 0$ true.

$x = 3$