Trigonometry Examples

$\log_{3} (3 x) = \log_{3} (x) + \log_{3} (4 - x)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $\log_{3} (x) + \log_{3} (4 - x)$ .

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Step 1.1.1

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

$\log_{3} (3 x) = \log_{3} (x (4 - x))$

Step 1.1.2

Simplify by multiplying through.

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Step 1.1.2.1

Apply the distributive property.

$\log_{3} (3 x) = \log_{3} (x \cdot 4 + x (- x))$

Step 1.1.2.2

Reorder.

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Step 1.1.2.2.1

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

$\log_{3} (3 x) = \log_{3} (4 \cdot x + x (- x))$

Step 1.1.2.2.2

Rewrite using the commutative property of multiplication.

$\log_{3} (3 x) = \log_{3} (4 \cdot x - x \cdot x)$

Step 1.1.3

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

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Step 1.1.3.1

Move $x$ .

$\log_{3} (3 x) = \log_{3} (4 x - (x \cdot x))$

Step 1.1.3.2

Multiply $x$ by $x$ .

$\log_{3} (3 x) = \log_{3} (4 x - x^{2})$

Step 2

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

$3 x = 4 x - x^{2}$

Step 3

Solve for $x$ .

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Step 3.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$4 x - x^{2} = 3 x$

Step 3.2

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

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Step 3.2.1

Subtract $3 x$ from both sides of the equation.

$4 x - x^{2} - 3 x = 0$

Step 3.2.2

Subtract $3 x$ from $4 x$ .

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

Step 3.3

Factor $- x$ out of $- x^{2} + x$ .

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Step 3.3.1

Factor $- x$ out of $- x^{2}$ .

$- x \cdot x + x = 0$

Step 3.3.2

Rewrite $x$ as $1 x$ .

$- x \cdot x + 1 x = 0$

Step 3.3.3

Factor $- x$ out of $1 x$ .

$- x \cdot x - x \cdot - 1 = 0$

Step 3.3.4

Factor $- x$ out of $- x (x) - x (- 1)$ .

$- x (x - 1) = 0$

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 $0$ .

$x = 0$

$x - 1 = 0$

Step 3.5

Set $x$ equal to $0$ .

$x = 0$

Step 3.6

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

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Step 3.6.1

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

$x - 1 = 0$

Step 3.6.2

Add $1$ to both sides of the equation.

$x = 1$

Step 3.7

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

$x = 0, 1$

Step 4

Exclude the solutions that do not make $\log_{3} (3 x) = \log_{3} (x) + \log_{3} (4 - x)$ true.

$x = 1$