Precalculus Examples

$\log_{a} (\frac{a}{\sqrt{x}}) - \log_{a} (a x)$

Step 1

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

$\log_{a} (\frac{\frac{a}{\sqrt{x}}}{a x})$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\log_{a} (\frac{a}{\sqrt{x}} \cdot \frac{1}{a x})$

Step 3

Cancel the common factor of $a$ .

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Factor $a$ out of $a x$ .

$\log_{a} (\frac{a}{\sqrt{x}} \cdot \frac{1}{a (x)})$

Cancel the common factor.

$\log_{a} (\frac{a}{\sqrt{x}} \cdot \frac{1}{a x})$

Rewrite the expression.

$\log_{a} (\frac{1}{\sqrt{x}} \cdot \frac{1}{x})$

Step 4

Multiply $\frac{1}{\sqrt{x}}$ by $\frac{1}{x}$ .

$\log_{a} (\frac{1}{\sqrt{x} x})$

Step 5

Multiply $\frac{1}{\sqrt{x} x}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\log_{a} (\frac{1}{\sqrt{x} x} \cdot \frac{\sqrt{x}}{\sqrt{x}})$

Step 6

Combine and simplify the denominator.

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Multiply $\frac{1}{\sqrt{x} x}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\log_{a} (\frac{\sqrt{x}}{\sqrt{x} x \sqrt{x}})$

Move $\sqrt{x}$ .

$\log_{a} (\frac{\sqrt{x}}{x (\sqrt{x} \sqrt{x})})$

Raise $\sqrt{x}$ to the power of $1$ .

$\log_{a} (\frac{\sqrt{x}}{x ({\sqrt{x}}^{1} \sqrt{x})})$

Raise $\sqrt{x}$ to the power of $1$ .

$\log_{a} (\frac{\sqrt{x}}{x ({\sqrt{x}}^{1} {\sqrt{x}}^{1})})$

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

$\log_{a} (\frac{\sqrt{x}}{x {\sqrt{x}}^{1 + 1}})$

Add $1$ and $1$ .

$\log_{a} (\frac{\sqrt{x}}{x {\sqrt{x}}^{2}})$

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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

$\log_{a} (\frac{\sqrt{x}}{x {(x^{\frac{1}{2}})}^{2}})$

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

$\log_{a} (\frac{\sqrt{x}}{x \cdot x^{\frac{1}{2} \cdot 2}})$

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

$\log_{a} (\frac{\sqrt{x}}{x \cdot x^{\frac{2}{2}}})$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\log_{a} (\frac{\sqrt{x}}{x \cdot x^{\frac{2}{2}}})$

Rewrite the expression.

$\log_{a} (\frac{\sqrt{x}}{x \cdot x^{1}})$

Simplify.

$\log_{a} (\frac{\sqrt{x}}{x \cdot x})$

Step 7

Multiply $x$ by $x$ .

$\log_{a} (\frac{\sqrt{x}}{x^{2}})$