Precalculus Examples

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

Step 1

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

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

Step 2

Expand $\log_{5} (x^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 3

Expand $\log_{5} (x^{7})$ by moving $7$ outside the logarithm.

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

Step 4

Simplify each term.

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Step 4.1

Combine $\frac{1}{2}$ and $\log_{5} (x)$ .

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

Step 4.2

Multiply $7$ by $- 1$ .

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

Step 5

To write $- 7 \log_{5} (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6

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

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

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Step 8.1

Factor $\log_{5} (x)$ out of $\log_{5} (x) - 7 \log_{5} (x) \cdot 2$ .

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Step 8.1.1

Multiply by $1$ .

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

Step 8.1.2

Factor $\log_{5} (x)$ out of $- 7 \log_{5} (x) \cdot 2$ .

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

Step 8.1.3

Factor $\log_{5} (x)$ out of $\log_{5} (x) \cdot 1 + \log_{5} (x) (- 7 \cdot 2)$ .

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

Step 8.2

Multiply $- 7$ by $2$ .

$\frac{\log_{5} (x) (1 - 14)}{2}$

Step 8.3

Subtract $14$ from $1$ .

$\frac{\log_{5} (x) \cdot - 13}{2}$

Step 9

Move $- 13$ to the left of $\log_{5} (x)$ .

$\frac{- 13 \cdot \log_{5} (x)}{2}$

Step 10

Move the negative in front of the fraction.

$- \frac{13 \log_{5} (x)}{2}$