Precalculus Examples

$\ln (\frac{10}{x}) - \ln (10 x)$

Step 1

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

$\ln (\frac{\frac{10}{x}}{10 x})$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{10}{x} \cdot \frac{1}{10 x})$

Step 3

Cancel the common factor of $10$ .

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Step 3.1

Factor $10$ out of $10 x$ .

$\ln (\frac{10}{x} \cdot \frac{1}{10 (x)})$

Step 3.2

Cancel the common factor.

$\ln (\frac{10}{x} \cdot \frac{1}{10 x})$

Step 3.3

Rewrite the expression.

$\ln (\frac{1}{x} \cdot \frac{1}{x})$

Step 4

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

$\ln (\frac{1}{x \cdot x})$

Step 5

Raise $x$ to the power of $1$ .

$\ln (\frac{1}{x^{1} x})$

Step 6

Raise $x$ to the power of $1$ .

$\ln (\frac{1}{x^{1} x^{1}})$

Step 7

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

$\ln (\frac{1}{x^{1 + 1}})$

Step 8

Add $1$ and $1$ .

$\ln (\frac{1}{x^{2}})$