Precalculus Examples

$\ln (x - 35) + \ln (x + 180) = 3 \ln (6)$

Step 1

Simplify the left side.

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Step 1.1

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

$\ln ((x - 35) (x + 180)) = 3 \ln (6)$

Step 1.2

Expand $(x - 35) (x + 180)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\ln (x (x + 180) - 35 (x + 180)) = 3 \ln (6)$

Step 1.2.2

Apply the distributive property.

$\ln (x \cdot x + x \cdot 180 - 35 (x + 180)) = 3 \ln (6)$

Step 1.2.3

Apply the distributive property.

$\ln (x \cdot x + x \cdot 180 - 35 x - 35 \cdot 180) = 3 \ln (6)$

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 $x$ .

$\ln (x^{2} + x \cdot 180 - 35 x - 35 \cdot 180) = 3 \ln (6)$

Step 1.3.1.2

Move $180$ to the left of $x$ .

$\ln (x^{2} + 180 \cdot x - 35 x - 35 \cdot 180) = 3 \ln (6)$

Step 1.3.1.3

Multiply $- 35$ by $180$ .

$\ln (x^{2} + 180 x - 35 x - 6300) = 3 \ln (6)$

Step 1.3.2

Subtract $35 x$ from $180 x$ .

$\ln (x^{2} + 145 x - 6300) = 3 \ln (6)$

Step 2

Simplify $3 \ln (6)$ by moving $3$ inside the logarithm.

$\ln (x^{2} + 145 x - 6300) = \ln (6^{3})$

Step 3

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

$x^{2} + 145 x - 6300 = 6^{3}$

Step 4

Solve for $x$ .

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Step 4.1

Raise $6$ to the power of $3$ .

$x^{2} + 145 x - 6300 = 216$

Step 4.2

Subtract $216$ from both sides of the equation.

$x^{2} + 145 x - 6300 - 216 = 0$

Step 4.3

Subtract $216$ from $- 6300$ .

$x^{2} + 145 x - 6516 = 0$

Step 4.4

Factor $x^{2} + 145 x - 6516$ using the AC method.

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Step 4.4.1

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 $- 6516$ and whose sum is $145$ .

$- 36, 181$

Step 4.4.2

Write the factored form using these integers.

$(x - 36) (x + 181) = 0$

Step 4.5

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

$x - 36 = 0$

$x + 181 = 0$

Step 4.6

Set $x - 36$ equal to $0$ and solve for $x$ .

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Step 4.6.1

Set $x - 36$ equal to $0$ .

$x - 36 = 0$

Step 4.6.2

Add $36$ to both sides of the equation.

$x = 36$

Step 4.7

Set $x + 181$ equal to $0$ and solve for $x$ .

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Step 4.7.1

Set $x + 181$ equal to $0$ .

$x + 181 = 0$

Step 4.7.2

Subtract $181$ from both sides of the equation.

$x = - 181$

Step 4.8

The final solution is all the values that make $(x - 36) (x + 181) = 0$ true.

$x = 36, - 181$

Step 5

Exclude the solutions that do not make $\ln (x - 35) + \ln (x + 180) = 3 \ln (6)$ true.

$x = 36$