Precalculus Examples

$\log_{b} (2 x^{2} + 5 x - 7) - \log_{b} (2 x + 7) (r^{2} - 1) + \log_{b} (x + 1) = 0$

Step 1

Simplify the left side.

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Step 1.1

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

$- \log_{b} (2 x + 7) (r^{2} - 1) + \log_{b} ((2 x^{2} + 5 x - 7) (x + 1)) = 0$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the distributive property.

$- \log_{b} (2 x + 7) r^{2} - \log_{b} (2 x + 7) \cdot - 1 + \log_{b} ((2 x^{2} + 5 x - 7) (x + 1)) = 0$

Step 1.2.2

Multiply $- \log_{b} (2 x + 7) \cdot - 1$ .

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Step 1.2.2.1

Multiply $- 1$ by $- 1$ .

$- \log_{b} (2 x + 7) r^{2} + 1 \log_{b} (2 x + 7) + \log_{b} ((2 x^{2} + 5 x - 7) (x + 1)) = 0$

Step 1.2.2.2

Multiply $\log_{b} (2 x + 7)$ by $1$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (2 x + 7) + \log_{b} ((2 x^{2} + 5 x - 7) (x + 1)) = 0$

Step 1.2.3

Expand $(2 x^{2} + 5 x - 7) (x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (2 x + 7) + \log_{b} (2 x^{2} x + 2 x^{2} \cdot 1 + 5 x \cdot x + 5 x \cdot 1 - 7 x - 7 \cdot 1) = 0$

Step 1.2.4

Simplify each term.

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Step 1.2.4.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.2.4.1.1

Move $x$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (2 x + 7) + \log_{b} (2 (x \cdot x^{2}) + 2 x^{2} \cdot 1 + 5 x \cdot x + 5 x \cdot 1 - 7 x - 7 \cdot 1) = 0$

Step 1.2.4.1.2

Multiply $x$ by $x^{2}$ .

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Step 1.2.4.1.2.1

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

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (2 x + 7) + \log_{b} (2 (x \cdot x^{2}) + 2 x^{2} \cdot 1 + 5 x \cdot x + 5 x \cdot 1 - 7 x - 7 \cdot 1) = 0$

Step 1.2.4.1.2.2

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

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (2 x + 7) + \log_{b} (2 x^{1 + 2} + 2 x^{2} \cdot 1 + 5 x \cdot x + 5 x \cdot 1 - 7 x - 7 \cdot 1) = 0$

Step 1.2.4.1.3

Add $1$ and $2$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (2 x + 7) + \log_{b} (2 x^{3} + 2 x^{2} \cdot 1 + 5 x \cdot x + 5 x \cdot 1 - 7 x - 7 \cdot 1) = 0$

Step 1.2.4.2

Multiply $2$ by $1$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (2 x + 7) + \log_{b} (2 x^{3} + 2 x^{2} + 5 x \cdot x + 5 x \cdot 1 - 7 x - 7 \cdot 1) = 0$

Step 1.2.4.3

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

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Step 1.2.4.3.1

Move $x$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (2 x + 7) + \log_{b} (2 x^{3} + 2 x^{2} + 5 (x \cdot x) + 5 x \cdot 1 - 7 x - 7 \cdot 1) = 0$

Step 1.2.4.3.2

Multiply $x$ by $x$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (2 x + 7) + \log_{b} (2 x^{3} + 2 x^{2} + 5 x^{2} + 5 x \cdot 1 - 7 x - 7 \cdot 1) = 0$

Step 1.2.4.4

Multiply $5$ by $1$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (2 x + 7) + \log_{b} (2 x^{3} + 2 x^{2} + 5 x^{2} + 5 x - 7 x - 7 \cdot 1) = 0$

Step 1.2.4.5

Multiply $- 7$ by $1$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (2 x + 7) + \log_{b} (2 x^{3} + 2 x^{2} + 5 x^{2} + 5 x - 7 x - 7) = 0$

Step 1.2.5

Add $2 x^{2}$ and $5 x^{2}$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (2 x + 7) + \log_{b} (2 x^{3} + 7 x^{2} + 5 x - 7 x - 7) = 0$

Step 1.2.6

Subtract $7 x$ from $5 x$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (2 x + 7) + \log_{b} (2 x^{3} + 7 x^{2} - 2 x - 7) = 0$

Step 1.3

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

$- \log_{b} (2 x + 7) r^{2} + \log_{b} ((2 x + 7) (2 x^{3} + 7 x^{2} - 2 x - 7)) = 0$

Step 1.4

Simplify each term.

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Step 1.4.1

Expand $(2 x + 7) (2 x^{3} + 7 x^{2} - 2 x - 7)$ by multiplying each term in the first expression by each term in the second expression.

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (2 x (2 x^{3}) + 2 x (7 x^{2}) + 2 x (- 2 x) + 2 x \cdot - 7 + 7 (2 x^{3}) + 7 (7 x^{2}) + 7 (- 2 x) + 7 \cdot - 7) = 0$

Step 1.4.2

Simplify each term.

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Step 1.4.2.1

Rewrite using the commutative property of multiplication.

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (2 \cdot (2 x \cdot x^{3}) + 2 x (7 x^{2}) + 2 x (- 2 x) + 2 x \cdot - 7 + 7 (2 x^{3}) + 7 (7 x^{2}) + 7 (- 2 x) + 7 \cdot - 7) = 0$

Step 1.4.2.2

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 1.4.2.2.1

Move $x^{3}$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (2 \cdot (2 (x^{3} x)) + 2 x (7 x^{2}) + 2 x (- 2 x) + 2 x \cdot - 7 + 7 (2 x^{3}) + 7 (7 x^{2}) + 7 (- 2 x) + 7 \cdot - 7) = 0$

Step 1.4.2.2.2

Multiply $x^{3}$ by $x$ .

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Step 1.4.2.2.2.1

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

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (2 \cdot (2 (x^{3} x)) + 2 x (7 x^{2}) + 2 x (- 2 x) + 2 x \cdot - 7 + 7 (2 x^{3}) + 7 (7 x^{2}) + 7 (- 2 x) + 7 \cdot - 7) = 0$

Step 1.4.2.2.2.2

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

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (2 \cdot (2 x^{3 + 1}) + 2 x (7 x^{2}) + 2 x (- 2 x) + 2 x \cdot - 7 + 7 (2 x^{3}) + 7 (7 x^{2}) + 7 (- 2 x) + 7 \cdot - 7) = 0$

Step 1.4.2.2.3

Add $3$ and $1$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (2 \cdot (2 x^{4}) + 2 x (7 x^{2}) + 2 x (- 2 x) + 2 x \cdot - 7 + 7 (2 x^{3}) + 7 (7 x^{2}) + 7 (- 2 x) + 7 \cdot - 7) = 0$

Step 1.4.2.3

Multiply $2$ by $2$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (4 x^{4} + 2 x (7 x^{2}) + 2 x (- 2 x) + 2 x \cdot - 7 + 7 (2 x^{3}) + 7 (7 x^{2}) + 7 (- 2 x) + 7 \cdot - 7) = 0$

Step 1.4.2.4

Rewrite using the commutative property of multiplication.

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (4 x^{4} + 2 \cdot (7 x \cdot x^{2}) + 2 x (- 2 x) + 2 x \cdot - 7 + 7 (2 x^{3}) + 7 (7 x^{2}) + 7 (- 2 x) + 7 \cdot - 7) = 0$

Step 1.4.2.5

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.4.2.5.1

Move $x^{2}$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (4 x^{4} + 2 \cdot (7 (x^{2} x)) + 2 x (- 2 x) + 2 x \cdot - 7 + 7 (2 x^{3}) + 7 (7 x^{2}) + 7 (- 2 x) + 7 \cdot - 7) = 0$

Step 1.4.2.5.2

Multiply $x^{2}$ by $x$ .

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Step 1.4.2.5.2.1

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

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (4 x^{4} + 2 \cdot (7 (x^{2} x)) + 2 x (- 2 x) + 2 x \cdot - 7 + 7 (2 x^{3}) + 7 (7 x^{2}) + 7 (- 2 x) + 7 \cdot - 7) = 0$

Step 1.4.2.5.2.2

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

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (4 x^{4} + 2 \cdot (7 x^{2 + 1}) + 2 x (- 2 x) + 2 x \cdot - 7 + 7 (2 x^{3}) + 7 (7 x^{2}) + 7 (- 2 x) + 7 \cdot - 7) = 0$

Step 1.4.2.5.3

Add $2$ and $1$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (4 x^{4} + 2 \cdot (7 x^{3}) + 2 x (- 2 x) + 2 x \cdot - 7 + 7 (2 x^{3}) + 7 (7 x^{2}) + 7 (- 2 x) + 7 \cdot - 7) = 0$

Step 1.4.2.6

Multiply $2$ by $7$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (4 x^{4} + 14 x^{3} + 2 x (- 2 x) + 2 x \cdot - 7 + 7 (2 x^{3}) + 7 (7 x^{2}) + 7 (- 2 x) + 7 \cdot - 7) = 0$

Step 1.4.2.7

Rewrite using the commutative property of multiplication.

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (4 x^{4} + 14 x^{3} + 2 \cdot (- 2 x \cdot x) + 2 x \cdot - 7 + 7 (2 x^{3}) + 7 (7 x^{2}) + 7 (- 2 x) + 7 \cdot - 7) = 0$

Step 1.4.2.8

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

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Step 1.4.2.8.1

Move $x$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (4 x^{4} + 14 x^{3} + 2 \cdot (- 2 (x \cdot x)) + 2 x \cdot - 7 + 7 (2 x^{3}) + 7 (7 x^{2}) + 7 (- 2 x) + 7 \cdot - 7) = 0$

Step 1.4.2.8.2

Multiply $x$ by $x$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (4 x^{4} + 14 x^{3} + 2 \cdot (- 2 x^{2}) + 2 x \cdot - 7 + 7 (2 x^{3}) + 7 (7 x^{2}) + 7 (- 2 x) + 7 \cdot - 7) = 0$

Step 1.4.2.9

Multiply $2$ by $- 2$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (4 x^{4} + 14 x^{3} - 4 x^{2} + 2 x \cdot - 7 + 7 (2 x^{3}) + 7 (7 x^{2}) + 7 (- 2 x) + 7 \cdot - 7) = 0$

Step 1.4.2.10

Multiply $- 7$ by $2$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (4 x^{4} + 14 x^{3} - 4 x^{2} - 14 x + 7 (2 x^{3}) + 7 (7 x^{2}) + 7 (- 2 x) + 7 \cdot - 7) = 0$

Step 1.4.2.11

Multiply $2$ by $7$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (4 x^{4} + 14 x^{3} - 4 x^{2} - 14 x + 14 x^{3} + 7 (7 x^{2}) + 7 (- 2 x) + 7 \cdot - 7) = 0$

Step 1.4.2.12

Multiply $7$ by $7$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (4 x^{4} + 14 x^{3} - 4 x^{2} - 14 x + 14 x^{3} + 49 x^{2} + 7 (- 2 x) + 7 \cdot - 7) = 0$

Step 1.4.2.13

Multiply $- 2$ by $7$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (4 x^{4} + 14 x^{3} - 4 x^{2} - 14 x + 14 x^{3} + 49 x^{2} - 14 x + 7 \cdot - 7) = 0$

Step 1.4.2.14

Multiply $7$ by $- 7$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (4 x^{4} + 14 x^{3} - 4 x^{2} - 14 x + 14 x^{3} + 49 x^{2} - 14 x - 49) = 0$

Step 1.4.3

Add $14 x^{3}$ and $14 x^{3}$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (4 x^{4} + 28 x^{3} - 4 x^{2} - 14 x + 49 x^{2} - 14 x - 49) = 0$

Step 1.4.4

Add $- 4 x^{2}$ and $49 x^{2}$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (4 x^{4} + 28 x^{3} + 45 x^{2} - 14 x - 14 x - 49) = 0$

Step 1.4.5

Subtract $14 x$ from $- 14 x$ .

$- \log_{b} (2 x + 7) r^{2} + \log_{b} (4 x^{4} + 28 x^{3} + 45 x^{2} - 28 x - 49) = 0$

Step 1.5

Reorder factors in $- \log_{b} (2 x + 7) r^{2} + \log_{b} (4 x^{4} + 28 x^{3} + 45 x^{2} - 28 x - 49)$ .

$- r^{2} \log_{b} (2 x + 7) + \log_{b} (4 x^{4} + 28 x^{3} + 45 x^{2} - 28 x - 49) = 0$

Step 2

Rewrite $\log_{b} (4 x^{4} + 28 x^{3} + 45 x^{2} - 28 x - 49) = r^{2} \log_{b} (2 x + 7)$ 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 $b^{y} = x$ .

$b^{r^{2} \log_{b} (2 x + 7)} = 4 x^{4} + 28 x^{3} + 45 x^{2} - 28 x - 49$

Step 3

Solve for $x$ .

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Step 3.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (b^{r^{2} \log_{b} (2 x + 7)}) = \ln (4 x^{4} + 28 x^{3} + 45 x^{2} - 28 x - 49)$

Step 3.2

Expand $\ln (b^{r^{2} \log_{b} (2 x + 7)})$ by moving $r^{2} \log_{b} (2 x + 7)$ outside the logarithm.

$r^{2} \log_{b} (2 x + 7) \ln (b) = \ln (4 x^{4} + 28 x^{3} + 45 x^{2} - 28 x - 49)$