Precalculus Examples

$8 \log_{5} (\sqrt{7 x - 2}) - \log_{5} (\frac{5}{x}) + \log_{5} (5)$

Step 1

Simplify each term.

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Step 1.1

Simplify $8 \log_{5} (\sqrt{7 x - 2})$ by moving $8$ inside the logarithm.

$\log_{5} ({\sqrt{7 x - 2}}^{8}) - \log_{5} (\frac{5}{x}) + \log_{5} (5)$

Step 1.2

Rewrite ${\sqrt{7 x - 2}}^{8}$ as ${(7 x - 2)}^{4}$ .

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Step 1.2.1

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

$\log_{5} ({({(7 x - 2)}^{\frac{1}{2}})}^{8}) - \log_{5} (\frac{5}{x}) + \log_{5} (5)$

Step 1.2.2

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

$\log_{5} ({(7 x - 2)}^{\frac{1}{2} \cdot 8}) - \log_{5} (\frac{5}{x}) + \log_{5} (5)$

Step 1.2.3

Combine $\frac{1}{2}$ and $8$ .

$\log_{5} ({(7 x - 2)}^{\frac{8}{2}}) - \log_{5} (\frac{5}{x}) + \log_{5} (5)$

Step 1.2.4

Cancel the common factor of $8$ and $2$ .

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Step 1.2.4.1

Factor $2$ out of $8$ .

$\log_{5} ({(7 x - 2)}^{\frac{2 \cdot 4}{2}}) - \log_{5} (\frac{5}{x}) + \log_{5} (5)$

Step 1.2.4.2

Cancel the common factors.

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Step 1.2.4.2.1

Factor $2$ out of $2$ .

$\log_{5} ({(7 x - 2)}^{\frac{2 \cdot 4}{2 (1)}}) - \log_{5} (\frac{5}{x}) + \log_{5} (5)$

Step 1.2.4.2.2

Cancel the common factor.

$\log_{5} ({(7 x - 2)}^{\frac{2 \cdot 4}{2 \cdot 1}}) - \log_{5} (\frac{5}{x}) + \log_{5} (5)$

Step 1.2.4.2.3

Rewrite the expression.

$\log_{5} ({(7 x - 2)}^{\frac{4}{1}}) - \log_{5} (\frac{5}{x}) + \log_{5} (5)$

Step 1.2.4.2.4

Divide $4$ by $1$ .

$\log_{5} ({(7 x - 2)}^{4}) - \log_{5} (\frac{5}{x}) + \log_{5} (5)$

Step 1.3

Logarithm base $5$ of $5$ is $1$ .

$\log_{5} ({(7 x - 2)}^{4}) - \log_{5} (\frac{5}{x}) + 1$

Step 2

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

$\log_{5} (\frac{{(7 x - 2)}^{4}}{\frac{5}{x}}) + 1$

Step 3

Simplify each term.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\log_{5} ({(7 x - 2)}^{4} \frac{x}{5}) + 1$

Step 3.2

Combine ${(7 x - 2)}^{4}$ and $\frac{x}{5}$ .

$\log_{5} (\frac{{(7 x - 2)}^{4} x}{5}) + 1$

Step 3.3

Rewrite $\frac{{(7 x - 2)}^{4} x}{5}$ as ${(7 x - 2)}^{4} x \cdot 5^{- 1}$ .

$\log_{5} ({(7 x - 2)}^{4} x \cdot 5^{- 1}) + 1$

Step 3.4

Rewrite $\log_{5} ({(7 x - 2)}^{4} x \cdot 5^{- 1})$ as $\log_{5} ({(7 x - 2)}^{4} x) + \log_{5} (5^{- 1})$ .

$\log_{5} ({(7 x - 2)}^{4} x) + \log_{5} (5^{- 1}) + 1$

Step 3.5

Use logarithm rules to move $- 1$ out of the exponent.

$\log_{5} ({(7 x - 2)}^{4} x) - \log_{5} (5) + 1$

Step 3.6

Logarithm base $5$ of $5$ is $1$ .

$\log_{5} ({(7 x - 2)}^{4} x) - 1 \cdot 1 + 1$

Step 3.7

Multiply $- 1$ by $1$ .

$\log_{5} ({(7 x - 2)}^{4} x) - 1 + 1$

Step 4

Simplify the expression.

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Step 4.1

Add $- 1$ and $1$ .

$\log_{5} ({(7 x - 2)}^{4} x) + 0$

Step 4.2

Add $\log_{5} ({(7 x - 2)}^{4} x)$ and $0$ .

$\log_{5} ({(7 x - 2)}^{4} x)$

Step 4.3

Reorder factors in $\log_{5} ({(7 x - 2)}^{4} x)$ .

$\log_{5} (x {(7 x - 2)}^{4})$