Trigonometry Examples

$\frac{3}{\frac{\log (6)}{\log (\frac{1}{8})}} - 4$

Step 1

Simplify each term.

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Step 1.1

Multiply the numerator by the reciprocal of the denominator.

$3 \frac{\log (\frac{1}{8})}{\log (6)} - 4$

Step 1.2

Multiply $3 \frac{\log (\frac{1}{8})}{\log (6)}$ .

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Step 1.2.1

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

$\frac{3 \log (\frac{1}{8})}{\log (6)} - 4$

Step 1.2.2

Simplify $3 \log (\frac{1}{8})$ by moving $3$ inside the logarithm.

$\frac{\log ({(\frac{1}{8})}^{3})}{\log (6)} - 4$

Step 1.3

Simplify the numerator.

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Step 1.3.1

Apply the product rule to $\frac{1}{8}$ .

$\frac{\log (\frac{1^{3}}{8^{3}})}{\log (6)} - 4$

Step 1.3.2

One to any power is one.

$\frac{\log (\frac{1}{8^{3}})}{\log (6)} - 4$

Step 1.3.3

Raise $8$ to the power of $3$ .

$\frac{\log (\frac{1}{512})}{\log (6)} - 4$

Step 2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{\log (6)}{\log (6)}$ .

$\frac{\log (\frac{1}{512})}{\log (6)} - 4 \cdot \frac{\log (6)}{\log (6)}$

Step 3

Combine fractions.

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Step 3.1

Combine $- 4$ and $\frac{\log (6)}{\log (6)}$ .

$\frac{\log (\frac{1}{512})}{\log (6)} + \frac{- 4 \log (6)}{\log (6)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{\log (\frac{1}{512}) - 4 \log (6)}{\log (6)}$

Step 4

Simplify the numerator.

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Step 4.1

Simplify $- 4 \log (6)$ by moving $4$ inside the logarithm.

$\frac{\log (\frac{1}{512}) - \log (6^{4})}{\log (6)}$

Step 4.2

Raise $6$ to the power of $4$ .

$\frac{\log (\frac{1}{512}) - \log (1296)}{\log (6)}$

Step 4.3

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

$\frac{\log (\frac{\frac{1}{512}}{1296})}{\log (6)}$

Step 4.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{\log (\frac{1}{512} \cdot \frac{1}{1296})}{\log (6)}$

Step 4.5

Multiply $\frac{1}{512} \cdot \frac{1}{1296}$ .

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Step 4.5.1

Multiply $\frac{1}{512}$ by $\frac{1}{1296}$ .

$\frac{\log (\frac{1}{512 \cdot 1296})}{\log (6)}$

Step 4.5.2

Multiply $512$ by $1296$ .

$\frac{\log (\frac{1}{663552})}{\log (6)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\frac{1}{663552})}{\log (6)}$

Decimal Form:

$- 7.48167526 \dots$