Trigonometry Examples

$\log (5 x) + \log (2 x + 3) = 1 + 2 \log (3 - x)$

Step 1

Simplify $\log (5 x) + \log (2 x + 3)$ .

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Step 1.1

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

$\log (5 x (2 x + 3)) = 1 + 2 \log (3 - x)$

Step 1.2

Simplify by multiplying through.

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Step 1.2.1

Apply the distributive property.

$\log (5 x (2 x) + 5 x \cdot 3) = 1 + 2 \log (3 - x)$

Step 1.2.2

Simplify the expression.

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Step 1.2.2.1

Rewrite using the commutative property of multiplication.

$\log (5 \cdot 2 x \cdot x + 5 x \cdot 3) = 1 + 2 \log (3 - x)$

Step 1.2.2.2

Multiply $3$ by $5$ .

$\log (5 \cdot 2 x \cdot x + 15 x) = 1 + 2 \log (3 - x)$

Step 1.3

Simplify each term.

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Step 1.3.1

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

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Step 1.3.1.1

Move $x$ .

$\log (5 \cdot 2 (x \cdot x) + 15 x) = 1 + 2 \log (3 - x)$

Step 1.3.1.2

Multiply $x$ by $x$ .

$\log (5 \cdot 2 x^{2} + 15 x) = 1 + 2 \log (3 - x)$

Step 1.3.2

Multiply $5$ by $2$ .

$\log (10 x^{2} + 15 x) = 1 + 2 \log (3 - x)$

Step 2

Simplify $2 \log (3 - x)$ by moving $2$ inside the logarithm.

$\log (10 x^{2} + 15 x) = 1 + \log ({(3 - x)}^{2})$

Step 3

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

$x = \frac{6}{5}$

Step 4