Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log_{7} (\sqrt{8 x - 1})$ by moving $2$ inside the logarithm.

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

Step 1.2

Rewrite ${\sqrt{8 x - 1}}^{2}$ as $8 x - 1$ .

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 x - 1}$ as ${(8 x - 1)}^{\frac{1}{2}}$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Cancel the common factor of $2$ .

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Step 1.2.4.1

Cancel the common factor.

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

Step 1.2.4.2

Rewrite the expression.

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

Step 1.2.5

Simplify.

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

Step 1.3

Logarithm base $7$ of $7$ is $1$ .

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

Step 2

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

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

Step 3

Simplify each term.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Multiply $8 x \frac{x}{7}$ .

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Step 3.3.1

Combine $8$ and $\frac{x}{7}$ .

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

Step 3.3.2

Combine $x$ and $\frac{8 x}{7}$ .

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

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

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

Step 3.3.6

Add $1$ and $1$ .

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

Step 3.4

Rewrite $- 1 \frac{x}{7}$ as $- \frac{x}{7}$ .

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