Precalculus Examples

$2 \log_{5} (\sqrt{6 x - 1}) - \log_{5} (\frac{6}{x}) + \log_{5} (6)$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log_{5} (\sqrt{6 x - 1})$ by moving $2$ inside the logarithm.

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

Step 1.2

Rewrite ${\sqrt{6 x - 1}}^{2}$ as $6 x - 1$ .

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x - 1}$ as ${(6 x - 1)}^{\frac{1}{2}}$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Cancel the common factor of $2$ .

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Step 1.2.4.1

Cancel the common factor.

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

Step 1.2.4.2

Rewrite the expression.

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

Step 1.2.5

Simplify.

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

Step 2

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

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

Step 3

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

$\log_{5} (\frac{6 x - 1}{\frac{6}{x}} \cdot 6)$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log_{5} ((6 x - 1) \frac{x}{6} \cdot 6)$

Step 5

Apply the distributive property.

$\log_{5} ((6 x \frac{x}{6} - 1 \frac{x}{6}) \cdot 6)$

Step 6

Cancel the common factor of $6$ .

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Step 6.1

Factor $6$ out of $6 x$ .

$\log_{5} ((6 (x) \frac{x}{6} - 1 \frac{x}{6}) \cdot 6)$

Step 6.2

Cancel the common factor.

$\log_{5} ((6 x \frac{x}{6} - 1 \frac{x}{6}) \cdot 6)$

Step 6.3

Rewrite the expression.

$\log_{5} ((x \cdot x - 1 \frac{x}{6}) \cdot 6)$

Step 7

Raise $x$ to the power of $1$ .

$\log_{5} ((x^{1} x - 1 \frac{x}{6}) \cdot 6)$

Step 8

Raise $x$ to the power of $1$ .

$\log_{5} ((x^{1} x^{1} - 1 \frac{x}{6}) \cdot 6)$

Step 9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 10

Simplify the expression.

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Step 10.1

Add $1$ and $1$ .

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

Step 10.2

Rewrite $- 1 \frac{x}{6}$ as $- \frac{x}{6}$ .

$\log_{5} ((x^{2} - \frac{x}{6}) \cdot 6)$

Step 11

Apply the distributive property.

$\log_{5} (x^{2} \cdot 6 - \frac{x}{6} \cdot 6)$

Step 12

Move $6$ to the left of $x^{2}$ .

$\log_{5} (6 \cdot x^{2} - \frac{x}{6} \cdot 6)$

Step 13

Cancel the common factor of $6$ .

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Step 13.1

Move the leading negative in $- \frac{x}{6}$ into the numerator.

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

Step 13.2

Cancel the common factor.

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

Step 13.3

Rewrite the expression.

$\log_{5} (6 \cdot x^{2} - x)$

$\log_{5} (6 x^{2} - x)$