Trigonometry Examples

$\log_{4} (2 x + 3 y) = \log_{4} (2 x) + \log_{4} (3 y)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $\log_{4} (2 x) + \log_{4} (3 y)$ .

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Step 1.1.1

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

$\log_{4} (2 x + 3 y) = \log_{4} (2 x (3 y))$

Step 1.1.2

Rewrite using the commutative property of multiplication.

$\log_{4} (2 x + 3 y) = \log_{4} (2 \cdot 3 x y)$

Step 1.1.3

Multiply $2$ by $3$ .

$\log_{4} (2 x + 3 y) = \log_{4} (6 x y)$

Step 2

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

$2 x + 3 y = 6 x y$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $6 x y$ from both sides of the equation.

$2 x + 3 y - 6 x y = 0$

Step 3.2

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

$2 x - 6 x y = - 3 y$

Step 3.3

Factor $2 x$ out of $2 x - 6 x y$ .

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Step 3.3.1

Factor $2 x$ out of $2 x$ .

$2 x (1) - 6 x y = - 3 y$

Step 3.3.2

Factor $2 x$ out of $- 6 x y$ .

$2 x (1) + 2 x (- 3 y) = - 3 y$

Step 3.3.3

Factor $2 x$ out of $2 x (1) + 2 x (- 3 y)$ .

$2 x (1 - 3 y) = - 3 y$

Step 3.4

Divide each term in $2 x (1 - 3 y) = - 3 y$ by $2 (1 - 3 y)$ and simplify.

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Step 3.4.1

Divide each term in $2 x (1 - 3 y) = - 3 y$ by $2 (1 - 3 y)$ .

$\frac{2 x (1 - 3 y)}{2 (1 - 3 y)} = \frac{- 3 y}{2 (1 - 3 y)}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{2 x (1 - 3 y)}{2 (1 - 3 y)} = \frac{- 3 y}{2 (1 - 3 y)}$

Step 3.4.2.1.2

Rewrite the expression.

$\frac{x (1 - 3 y)}{1 - 3 y} = \frac{- 3 y}{2 (1 - 3 y)}$

Step 3.4.2.2

Cancel the common factor of $1 - 3 y$ .

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Step 3.4.2.2.1

Cancel the common factor.

$\frac{x (1 - 3 y)}{1 - 3 y} = \frac{- 3 y}{2 (1 - 3 y)}$

Step 3.4.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 3 y}{2 (1 - 3 y)}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Move the negative in front of the fraction.

$x = - \frac{3 y}{2 (1 - 3 y)}$