Calculus Examples

$\log (x + 3) + \log (x + 4) = \log (20)$

Step 1

Simplify the left side.

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Step 1.1

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

$\log ((x + 3) (x + 4)) = \log (20)$

Step 1.2

Expand $(x + 3) (x + 4)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\log (x (x + 4) + 3 (x + 4)) = \log (20)$

Step 1.2.2

Apply the distributive property.

$\log (x \cdot x + x \cdot 4 + 3 (x + 4)) = \log (20)$

Step 1.2.3

Apply the distributive property.

$\log (x \cdot x + x \cdot 4 + 3 x + 3 \cdot 4) = \log (20)$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

$\log (x^{2} + x \cdot 4 + 3 x + 3 \cdot 4) = \log (20)$

Step 1.3.1.2

Move $4$ to the left of $x$ .

$\log (x^{2} + 4 \cdot x + 3 x + 3 \cdot 4) = \log (20)$

Step 1.3.1.3

Multiply $3$ by $4$ .

$\log (x^{2} + 4 x + 3 x + 12) = \log (20)$

Step 1.3.2

Add $4 x$ and $3 x$ .

$\log (x^{2} + 7 x + 12) = \log (20)$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$x^{2} + 7 x + 12 = 20$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $20$ from both sides of the equation.

$x^{2} + 7 x + 12 - 20 = 0$

Step 3.2

Subtract $20$ from $12$ .

$x^{2} + 7 x - 8 = 0$

Step 3.3

Factor $x^{2} + 7 x - 8$ using the AC method.

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Step 3.3.1

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 $- 8$ and whose sum is $7$ .

$- 1, 8$

Step 3.3.2

Write the factored form using these integers.

$(x - 1) (x + 8) = 0$

Step 3.4

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

$x - 1 = 0$

$x + 8 = 0$

Step 3.5

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 3.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 3.6

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

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Step 3.6.1

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

$x + 8 = 0$

Step 3.6.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 3.7

The final solution is all the values that make $(x - 1) (x + 8) = 0$ true.

$x = 1, - 8$

Step 4

Exclude the solutions that do not make $\log (x + 3) + \log (x + 4) = \log (20)$ true.

$x = 1$