Precalculus Examples

$\log (\sqrt{x}) + 4 \log (x) - \log (7 x^{2})$

Step 1

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

$4 \log (x) + \log (\frac{\sqrt{x}}{7 x^{2}})$

Step 2

Simplify $4 \log (x)$ by moving $4$ inside the logarithm.

$\log (x^{4}) + \log (\frac{\sqrt{x}}{7 x^{2}})$

Step 3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (x^{4} \frac{\sqrt{x}}{7 x^{2}})$

Step 4

Cancel the common factor of $x^{2}$ .

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Step 4.1

Factor $x^{2}$ out of $x^{4}$ .

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

Step 4.2

Factor $x^{2}$ out of $7 x^{2}$ .

$\log (x^{2} x^{2} \frac{\sqrt{x}}{x^{2} \cdot 7})$

Step 4.3

Cancel the common factor.

$\log (x^{2} x^{2} \frac{\sqrt{x}}{x^{2} \cdot 7})$

Step 4.4

Rewrite the expression.

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

Step 5

Combine $x^{2}$ and $\frac{\sqrt{x}}{7}$ .

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