Trigonometry Examples

$\log_{3} (y) + \log_{3} (3) = 3$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{3} (y \cdot 3) = 3$

Step 1.2

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

$\log_{3} (3 y) = 3$

Step 2

Rewrite $\log_{3} (3 y) = 3$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$3^{3} = 3 y$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $3 y = 3^{3}$ .

$3 y = 3^{3}$

Step 3.2

Divide each term in $3 y = 3^{3}$ by $3$ and simplify.

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Step 3.2.1

Divide each term in $3 y = 3^{3}$ by $3$ .

$\frac{3 y}{3} = \frac{3^{3}}{3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{3^{3}}{3}$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{3^{3}}{3}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $3^{3}$ and $3$ .

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Step 3.2.3.1.1

Factor $3$ out of $3^{3}$ .

$y = \frac{3 \cdot 3^{2}}{3}$

Step 3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.1.2.1

Factor $3$ out of $3$ .

$y = \frac{3 \cdot 3^{2}}{3 (1)}$

Step 3.2.3.1.2.2

Cancel the common factor.

$y = \frac{3 \cdot 3^{2}}{3 \cdot 1}$

Step 3.2.3.1.2.3

Rewrite the expression.

$y = \frac{3^{2}}{1}$

Step 3.2.3.1.2.4

Divide $3^{2}$ by $1$ .

$y = 3^{2}$

Step 3.2.3.2

Raise $3$ to the power of $2$ .

$y = 9$