Precalculus Examples

$\ln (5 x + 2) = \ln (4) + \ln (x + 3)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $\ln (4) + \ln (x + 3)$ .

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Step 1.1.1

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

$\ln (5 x + 2) = \ln (4 (x + 3))$

Step 1.1.2

Apply the distributive property.

$\ln (5 x + 2) = \ln (4 x + 4 \cdot 3)$

Step 1.1.3

Multiply $4$ by $3$ .

$\ln (5 x + 2) = \ln (4 x + 12)$

Step 2

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

$5 x + 2 = 4 x + 12$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $4 x$ from both sides of the equation.

$5 x + 2 - 4 x = 12$

Step 3.1.2

Subtract $4 x$ from $5 x$ .

$x + 2 = 12$

Step 3.2

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

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Step 3.2.1

Subtract $2$ from both sides of the equation.

$x = 12 - 2$

Step 3.2.2

Subtract $2$ from $12$ .

$x = 10$