Precalculus Examples

$6 \log_{7} (\sqrt{6 x - 7}) - \log_{7} (\frac{4}{x}) + \log_{7} (4)$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x - 7}$ as ${(6 x - 7)}^{\frac{1}{2}}$ .

$6 \log_{7} ({(6 x - 7)}^{\frac{1}{2}}) - \log_{7} (\frac{4}{x}) + \log_{7} (4)$

Step 2

Expand $\log_{7} ({(6 x - 7)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$6 (\frac{1}{2} \log_{7} (6 x - 7)) - \log_{7} (\frac{4}{x}) + \log_{7} (4)$

Step 3

Rewrite $\log_{7} (\frac{4}{x})$ as $\log_{7} (4) - \log_{7} (x)$ .

$6 (\frac{1}{2} \log_{7} (6 x - 7)) - (\log_{7} (4) - \log_{7} (x)) + \log_{7} (4)$

Step 4

Simplify each term.

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Step 4.1

Combine $\frac{1}{2}$ and $\log_{7} (6 x - 7)$ .

$6 \frac{\log_{7} (6 x - 7)}{2} - (\log_{7} (4) - \log_{7} (x)) + \log_{7} (4)$

Step 4.2

Cancel the common factor of $2$ .

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Step 4.2.1

Factor $2$ out of $6$ .

$2 (3) \frac{\log_{7} (6 x - 7)}{2} - (\log_{7} (4) - \log_{7} (x)) + \log_{7} (4)$

Step 4.2.2

Cancel the common factor.

$2 \cdot 3 \frac{\log_{7} (6 x - 7)}{2} - (\log_{7} (4) - \log_{7} (x)) + \log_{7} (4)$

Step 4.2.3

Rewrite the expression.

$3 \log_{7} (6 x - 7) - (\log_{7} (4) - \log_{7} (x)) + \log_{7} (4)$

Step 4.3

Simplify each term.

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Step 4.3.1

Rewrite $\log_{7} (4)$ as $\log_{7} (2^{2})$ .

$3 \log_{7} (6 x - 7) - (\log_{7} (2^{2}) - \log_{7} (x)) + \log_{7} (4)$

Step 4.3.2

Expand $\log_{7} (2^{2})$ by moving $2$ outside the logarithm.

$3 \log_{7} (6 x - 7) - (2 \log_{7} (2) - \log_{7} (x)) + \log_{7} (4)$

Step 4.4

Apply the distributive property.

$3 \log_{7} (6 x - 7) - (2 \log_{7} (2)) - - \log_{7} (x) + \log_{7} (4)$

Step 4.5

Multiply $2$ by $- 1$ .

$3 \log_{7} (6 x - 7) - 2 \log_{7} (2) - - \log_{7} (x) + \log_{7} (4)$

Step 4.6

Multiply $- - \log_{7} (x)$ .

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Step 4.6.1

Multiply $- 1$ by $- 1$ .

$3 \log_{7} (6 x - 7) - 2 \log_{7} (2) + 1 \log_{7} (x) + \log_{7} (4)$

Step 4.6.2

Multiply $\log_{7} (x)$ by $1$ .

$3 \log_{7} (6 x - 7) - 2 \log_{7} (2) + \log_{7} (x) + \log_{7} (4)$

Step 4.7

Rewrite $\log_{7} (4)$ as $\log_{7} (2^{2})$ .

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

Step 4.8

Expand $\log_{7} (2^{2})$ by moving $2$ outside the logarithm.

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

Step 5

Add $- 2 \log_{7} (2)$ and $2 \log_{7} (2)$ .

$3 \log_{7} (6 x - 7) + 0 + \log_{7} (x)$

Step 6

Add $3 \log_{7} (6 x - 7)$ and $0$ .

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