Finite Math Examples

$\ln (k + 5) + \ln (k + 2) = \ln (18 k)$

Step 1

Simplify $\ln (k + 5) + \ln (k + 2)$ .

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Step 1.1

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

$\ln ((k + 5) (k + 2)) = \ln (18 k)$

Step 1.2

Expand $(k + 5) (k + 2)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\ln (k (k + 2) + 5 (k + 2)) = \ln (18 k)$

Step 1.2.2

Apply the distributive property.

$\ln (k \cdot k + k \cdot 2 + 5 (k + 2)) = \ln (18 k)$

Step 1.2.3

Apply the distributive property.

$\ln (k \cdot k + k \cdot 2 + 5 k + 5 \cdot 2) = \ln (18 k)$

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 $k$ by $k$ .

$\ln (k^{2} + k \cdot 2 + 5 k + 5 \cdot 2) = \ln (18 k)$

Step 1.3.1.2

Move $2$ to the left of $k$ .

$\ln (k^{2} + 2 \cdot k + 5 k + 5 \cdot 2) = \ln (18 k)$

Step 1.3.1.3

Multiply $5$ by $2$ .

$\ln (k^{2} + 2 k + 5 k + 10) = \ln (18 k)$

Step 1.3.2

Add $2 k$ and $5 k$ .

$\ln (k^{2} + 7 k + 10) = \ln (18 k)$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$k = 1, 10$

Step 3