Trigonometry Examples

$\log_{12} (2 x + 1) = 1 - 2 \log_{12} (2)$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$\log_{12} (2 x + 1) + 2 \log_{12} (2) = 1$

Step 2

Simplify the left side.

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Step 2.1

Simplify $\log_{12} (2 x + 1) + 2 \log_{12} (2)$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Simplify $2 \log_{12} (2)$ by moving $2$ inside the logarithm.

$\log_{12} (2 x + 1) + \log_{12} (2^{2}) = 1$

Step 2.1.1.2

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

$\log_{12} (2 x + 1) + \log_{12} (4) = 1$

Step 2.1.2

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

$\log_{12} ((2 x + 1) \cdot 4) = 1$

Step 2.1.3

Apply the distributive property.

$\log_{12} (2 x \cdot 4 + 1 \cdot 4) = 1$

Step 2.1.4

Multiply.

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Step 2.1.4.1

Multiply $4$ by $2$ .

$\log_{12} (8 x + 1 \cdot 4) = 1$

Step 2.1.4.2

Multiply $4$ by $1$ .

$\log_{12} (8 x + 4) = 1$

Step 3

Rewrite $\log_{12} (8 x + 4) = 1$ 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 $b^{y} = x$ .

$12^{1} = 8 x + 4$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $8 x + 4 = 12^{1}$ .

$8 x + 4 = 12$

Step 4.2

Move all terms not containing $x$ to the right side of the equation.

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Step 4.2.1

Subtract $4$ from both sides of the equation.

$8 x = 12 - 4$

Step 4.2.2

Subtract $4$ from $12$ .

$8 x = 8$

Step 4.3

Divide each term in $8 x = 8$ by $8$ and simplify.

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Step 4.3.1

Divide each term in $8 x = 8$ by $8$ .

$\frac{8 x}{8} = \frac{8}{8}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $8$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{8}{8}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{8}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $8$ by $8$ .

$x = 1$