Trigonometry Examples

$\log_{4} (y) = \log_{4} (x) - \log_{4} (5) + \log_{4} (3)$

Step 1

Rewrite the equation as $\log_{4} (x) - \log_{4} (5) + \log_{4} (3) = \log_{4} (y)$ .

$\log_{4} (x) - \log_{4} (5) + \log_{4} (3) = \log_{4} (y)$

Step 2

Simplify the left side.

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Step 2.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{4} (\frac{x}{5}) + \log_{4} (3) = \log_{4} (y)$

Step 2.2

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

$\log_{4} (\frac{x}{5} \cdot 3) = \log_{4} (y)$

Step 2.3

Combine $\frac{x}{5}$ and $3$ .

$\log_{4} (\frac{x \cdot 3}{5}) = \log_{4} (y)$

Step 2.4

Move $3$ to the left of $x$ .

$\log_{4} (\frac{3 x}{5}) = \log_{4} (y)$

Step 3

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

$\frac{3 x}{5} = y$

Step 4

Solve for $x$ .

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Step 4.1

Multiply both sides of the equation by $\frac{5}{3}$ .

$\frac{5}{3} \cdot \frac{3 x}{5} = \frac{5}{3} y$

Step 4.2

Simplify both sides of the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Simplify $\frac{5}{3} \cdot \frac{3 x}{5}$ .

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Step 4.2.1.1.1

Cancel the common factor of $5$ .

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Step 4.2.1.1.1.1

Cancel the common factor.

$\frac{5}{3} \cdot \frac{3 x}{5} = \frac{5}{3} y$

Step 4.2.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{5}{3} y$

Step 4.2.1.1.2

Cancel the common factor of $3$ .

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Step 4.2.1.1.2.1

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{5}{3} y$

Step 4.2.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{5}{3} y$

Step 4.2.1.1.2.3

Rewrite the expression.

$x = \frac{5}{3} y$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Combine $\frac{5}{3}$ and $y$ .

$x = \frac{5 y}{3}$