Trigonometry Examples

$2 \log_{6} (4) - \frac{1}{3} \cdot \log_{6} (8) = \log_{6} (x)$

Step 1

Rewrite the equation as $\log_{6} (x) = 2 \log_{6} (4) - \frac{1}{3} \cdot \log_{6} (8)$ .

$\log_{6} (x) = 2 \log_{6} (4) - \frac{1}{3} \cdot \log_{6} (8)$

Step 2

Simplify the right side.

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Step 2.1

Combine $\log_{6} (8)$ and $\frac{1}{3}$ .

$\log_{6} (x) = 2 \log_{6} (4) - \frac{\log_{6} (8)}{3}$

Step 3

Simplify the right side.

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Step 3.1

Simplify $2 \log_{6} (4) - \frac{\log_{6} (8)}{3}$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Simplify $2 \log_{6} (4)$ by moving $2$ inside the logarithm.

$\log_{6} (x) = \log_{6} (4^{2}) - \frac{\log_{6} (8)}{3}$

Step 3.1.1.2

Raise $4$ to the power of $2$ .

$\log_{6} (x) = \log_{6} (16) - \frac{\log_{6} (8)}{3}$

Step 3.1.1.3

Rewrite $\log_{6} (8)$ as $\log_{6} (2^{3})$ .

$\log_{6} (x) = \log_{6} (16) - \frac{\log_{6} (2^{3})}{3}$

Step 3.1.1.4

Expand $\log_{6} (2^{3})$ by moving $3$ outside the logarithm.

$\log_{6} (x) = \log_{6} (16) - \frac{3 \log_{6} (2)}{3}$

Step 3.1.1.5

Cancel the common factor of $3$ .

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Step 3.1.1.5.1

Cancel the common factor.

$\log_{6} (x) = \log_{6} (16) - \frac{3 \log_{6} (2)}{3}$

Step 3.1.1.5.2

Divide $\log_{6} (2)$ by $1$ .

$\log_{6} (x) = \log_{6} (16) - \log_{6} (2)$

Step 3.1.2

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

$\log_{6} (x) = \log_{6} (\frac{16}{2})$

Step 3.1.3

Divide $16$ by $2$ .

$\log_{6} (x) = \log_{6} (8)$

Step 4

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

$x = 8$