Calculus Examples

$\ln (5 x^{2} + 2 x - 15) - 2 \ln (2 x - 1) = 0$

Step 1

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

$\ln (5 x^{2} + 2 x - 15) - 2 \ln (2 x - 1) = 0$

Step 2

Simplify the left side.

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Step 2.1

Simplify $\ln (5 x^{2} + 2 x - 15) - 2 \ln (2 x - 1)$ .

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Step 2.1.1

Simplify $- 2 \ln (2 x - 1)$ by moving $2$ inside the logarithm.

$\ln (5 x^{2} + 2 x - 15) - \ln ({(2 x - 1)}^{2}) = 0$

Step 2.1.2

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

$\ln (\frac{5 x^{2} + 2 x - 15}{{(2 x - 1)}^{2}}) = 0$

Step 3

Rewrite $\ln (\frac{5 x^{2} + 2 x - 15}{{(2 x - 1)}^{2}}) = 0$ 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 $b^{y} = x$ .

$e^{0} = \frac{5 x^{2} + 2 x - 15}{{(2 x - 1)}^{2}}$

Step 4

Cross multiply to remove the fraction.

$5 x^{2} + 2 x - 15 = e^{0} ({(2 x - 1)}^{2})$

Step 5

Simplify $e^{0} ({(2 x - 1)}^{2})$ .

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Step 5.1

Simplify the expression.

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Step 5.1.1

Anything raised to $0$ is $1$ .

$5 x^{2} + 2 x - 15 = 1 {(2 x - 1)}^{2}$

Step 5.1.2

Multiply ${(2 x - 1)}^{2}$ by $1$ .

$5 x^{2} + 2 x - 15 = {(2 x - 1)}^{2}$

Step 5.1.3

Rewrite ${(2 x - 1)}^{2}$ as $(2 x - 1) (2 x - 1)$ .

$5 x^{2} + 2 x - 15 = (2 x - 1) (2 x - 1)$

Step 5.2

Expand $(2 x - 1) (2 x - 1)$ using the FOIL Method.

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Step 5.2.1

Apply the distributive property.

$5 x^{2} + 2 x - 15 = 2 x (2 x - 1) - 1 (2 x - 1)$

Step 5.2.2

Apply the distributive property.

$5 x^{2} + 2 x - 15 = 2 x (2 x) + 2 x \cdot - 1 - 1 (2 x - 1)$

Step 5.2.3

Apply the distributive property.

$5 x^{2} + 2 x - 15 = 2 x (2 x) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1$

Step 5.3

Simplify and combine like terms.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Rewrite using the commutative property of multiplication.

$5 x^{2} + 2 x - 15 = 2 \cdot 2 x \cdot x + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1$

Step 5.3.1.2

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

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Step 5.3.1.2.1

Move $x$ .

$5 x^{2} + 2 x - 15 = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1$

Step 5.3.1.2.2

Multiply $x$ by $x$ .

$5 x^{2} + 2 x - 15 = 2 \cdot 2 x^{2} + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1$

Step 5.3.1.3

Multiply $2$ by $2$ .

$5 x^{2} + 2 x - 15 = 4 x^{2} + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1$

Step 5.3.1.4

Multiply $- 1$ by $2$ .

$5 x^{2} + 2 x - 15 = 4 x^{2} - 2 x - 1 (2 x) - 1 \cdot - 1$

Step 5.3.1.5

Multiply $2$ by $- 1$ .

$5 x^{2} + 2 x - 15 = 4 x^{2} - 2 x - 2 x - 1 \cdot - 1$

Step 5.3.1.6

Multiply $- 1$ by $- 1$ .

$5 x^{2} + 2 x - 15 = 4 x^{2} - 2 x - 2 x + 1$

Step 5.3.2

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

$5 x^{2} + 2 x - 15 = 4 x^{2} - 4 x + 1$

Step 6

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

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Step 6.1

Subtract $4 x^{2}$ from both sides of the equation.

$5 x^{2} + 2 x - 15 - 4 x^{2} = - 4 x + 1$

Step 6.2

Add $4 x$ to both sides of the equation.

$5 x^{2} + 2 x - 15 - 4 x^{2} + 4 x = 1$

Step 6.3

Subtract $4 x^{2}$ from $5 x^{2}$ .

$x^{2} + 2 x - 15 + 4 x = 1$

Step 6.4

Add $2 x$ and $4 x$ .

$x^{2} + 6 x - 15 = 1$

Step 7

Move all terms not containing $x$ to the right side of the equation.

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Step 7.1

Add $15$ to both sides of the equation.

$x^{2} + 6 x = 1 + 15$

Step 7.2

Add $1$ and $15$ .

$x^{2} + 6 x = 16$

Step 8

Subtract $16$ from both sides of the equation.

$x^{2} + 6 x - 16 = 0$

Step 9

Factor $x^{2} + 6 x - 16$ using the AC method.

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Step 9.1

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 $- 16$ and whose sum is $6$ .

$- 2, 8$

Step 9.2

Write the factored form using these integers.

$(x - 2) (x + 8) = 0$

Step 10

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

$x - 2 = 0$

$x + 8 = 0$

Step 11

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 11.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 11.2

Add $2$ to both sides of the equation.

$x = 2$

Step 12

Set $x + 8$ equal to $0$ and solve for $x$ .

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Step 12.1

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Step 12.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 13

The final solution is all the values that make $(x - 2) (x + 8) = 0$ true.

$x = 2, - 8$

Step 14

Exclude the solutions that do not make $\ln (5 x^{2} + 2 x - 15) - 2 \ln (2 x - 1) = 0$ true.

$x = 2$