Precalculus Examples

$2 \log_{3} (\sqrt{7 x - 3}) - \log_{3} (\frac{5}{6}) + \log_{3} (5)$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log_{3} (\sqrt{7 x - 3})$ by moving $2$ inside the logarithm.

$\log_{3} ({\sqrt{7 x - 3}}^{2}) - \log_{3} (\frac{5}{6}) + \log_{3} (5)$

Step 1.2

Rewrite ${\sqrt{7 x - 3}}^{2}$ as $7 x - 3$ .

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 x - 3}$ as ${(7 x - 3)}^{\frac{1}{2}}$ .

$\log_{3} ({({(7 x - 3)}^{\frac{1}{2}})}^{2}) - \log_{3} (\frac{5}{6}) + \log_{3} (5)$

Step 1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{3} ({(7 x - 3)}^{\frac{1}{2} \cdot 2}) - \log_{3} (\frac{5}{6}) + \log_{3} (5)$

Step 1.2.3

Combine $\frac{1}{2}$ and $2$ .

$\log_{3} ({(7 x - 3)}^{\frac{2}{2}}) - \log_{3} (\frac{5}{6}) + \log_{3} (5)$

Step 1.2.4

Cancel the common factor of $2$ .

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Step 1.2.4.1

Cancel the common factor.

$\log_{3} ({(7 x - 3)}^{\frac{2}{2}}) - \log_{3} (\frac{5}{6}) + \log_{3} (5)$

Step 1.2.4.2

Rewrite the expression.

$\log_{3} ({(7 x - 3)}^{1}) - \log_{3} (\frac{5}{6}) + \log_{3} (5)$

Step 1.2.5

Simplify.

$\log_{3} (7 x - 3) - \log_{3} (\frac{5}{6}) + \log_{3} (5)$

Step 2

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

$\log_{3} (\frac{7 x - 3}{\frac{5}{6}}) + \log_{3} (5)$

Step 3

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

$\log_{3} (\frac{7 x - 3}{\frac{5}{6}} \cdot 5)$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log_{3} ((7 x - 3) \frac{6}{5} \cdot 5)$

Step 5

Apply the distributive property.

$\log_{3} ((7 x \frac{6}{5} - 3 (\frac{6}{5})) \cdot 5)$

Step 6

Multiply $7 x \frac{6}{5}$ .

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Step 6.1

Combine $7$ and $\frac{6}{5}$ .

$\log_{3} ((x \frac{7 \cdot 6}{5} - 3 (\frac{6}{5})) \cdot 5)$

Step 6.2

Multiply $7$ by $6$ .

$\log_{3} ((x \frac{42}{5} - 3 (\frac{6}{5})) \cdot 5)$

Step 6.3

Combine $x$ and $\frac{42}{5}$ .

$\log_{3} ((\frac{x \cdot 42}{5} - 3 (\frac{6}{5})) \cdot 5)$

Step 7

Multiply $- 3 (\frac{6}{5})$ .

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Step 7.1

Combine $- 3$ and $\frac{6}{5}$ .

$\log_{3} ((\frac{x \cdot 42}{5} + \frac{- 3 \cdot 6}{5}) \cdot 5)$

Step 7.2

Multiply $- 3$ by $6$ .

$\log_{3} ((\frac{x \cdot 42}{5} + \frac{- 18}{5}) \cdot 5)$

Step 8

Simplify each term.

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Step 8.1

Move $42$ to the left of $x$ .

$\log_{3} ((\frac{42 \cdot x}{5} + \frac{- 18}{5}) \cdot 5)$

Step 8.2

Move the negative in front of the fraction.

$\log_{3} ((\frac{42 x}{5} - \frac{18}{5}) \cdot 5)$

Step 9

Simplify terms.

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Step 9.1

Apply the distributive property.

$\log_{3} (\frac{42 x}{5} \cdot 5 - \frac{18}{5} \cdot 5)$

Step 9.2

Cancel the common factor of $5$ .

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Step 9.2.1

Cancel the common factor.

$\log_{3} (\frac{42 x}{5} \cdot 5 - \frac{18}{5} \cdot 5)$

Step 9.2.2

Rewrite the expression.

$\log_{3} (42 x - \frac{18}{5} \cdot 5)$

Step 9.3

Cancel the common factor of $5$ .

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Step 9.3.1

Move the leading negative in $- \frac{18}{5}$ into the numerator.

$\log_{3} (42 x + \frac{- 18}{5} \cdot 5)$

Step 9.3.2

Cancel the common factor.

$\log_{3} (42 x + \frac{- 18}{5} \cdot 5)$

Step 9.3.3

Rewrite the expression.

$\log_{3} (42 x - 18)$