Precalculus Examples

$\log (\frac{{(x - 7)}^{2} \sqrt{x}}{(5^{x}) \ln (x)})$

Step 1

Rewrite $\log (\frac{{(x - 7)}^{2} \sqrt{x}}{(5^{x}) \ln (x)})$ as $\log ({(x - 7)}^{2} \sqrt{x}) - \log ((5^{x}) \ln (x))$ .

$\log ({(x - 7)}^{2} \sqrt{x}) - \log ((5^{x}) \ln (x))$

Step 2

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

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

Step 3

Rewrite $\log ({(x - 7)}^{2} x^{\frac{1}{2}})$ as $\log ({(x - 7)}^{2}) + \log (x^{\frac{1}{2}})$ .

$\log ({(x - 7)}^{2}) + \log (x^{\frac{1}{2}}) - \log ((5^{x}) \ln (x))$

Step 4

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

$2 \log (x - 7) + \log (x^{\frac{1}{2}}) - \log ((5^{x}) \ln (x))$

Step 5

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

$2 \log (x - 7) + \frac{1}{2} \log (x) - \log ((5^{x}) \ln (x))$

Step 6

Rewrite $\log ((5^{x}) \ln (x))$ as $\log (5^{x}) + \log (\ln (x))$ .

$2 \log (x - 7) + \frac{1}{2} \log (x) - (\log (5^{x}) + \log (\ln (x)))$

Step 7

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

$2 \log (x - 7) + \frac{1}{2} \log (x) - (x \log (5) + \log (\ln (x)))$

Step 8

Simplify each term.

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Step 8.1

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

$2 \log (x - 7) + \frac{\log (x)}{2} - (x \log (5) + \log (\ln (x)))$

Step 8.2

Apply the distributive property.

$2 \log (x - 7) + \frac{\log (x)}{2} - x \log (5) - \log (\ln (x))$

Step 9

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

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

Step 10

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

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

Step 11

Combine the numerators over the common denominator.

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

Step 12

Multiply $2$ by $2$ .

$\frac{4 \log (x - 7) + \log (x)}{2} - x \log (5) - \log (\ln (x))$