Precalculus Examples

$(\frac{\ln (15) - \ln (3)}{\ln (75) - \ln (3)}) + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 1

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

$\frac{\ln (\frac{15}{3})}{\ln (75) - \ln (3)} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 2

Simplify each term.

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Step 2.1

Cancel the common factor of $15$ and $3$ .

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Step 2.1.1

Factor $3$ out of $15$ .

$\frac{\ln (\frac{3 \cdot 5}{3})}{\ln (75) - \ln (3)} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $3$ out of $3$ .

$\frac{\ln (\frac{3 \cdot 5}{3 (1)})}{\ln (75) - \ln (3)} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 2.1.2.2

Cancel the common factor.

$\frac{\ln (\frac{3 \cdot 5}{3 \cdot 1})}{\ln (75) - \ln (3)} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 2.1.2.3

Rewrite the expression.

$\frac{\ln (\frac{5}{1})}{\ln (75) - \ln (3)} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 2.1.2.4

Divide $5$ by $1$ .

$\frac{\ln (5)}{\ln (75) - \ln (3)} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 2.2

Simplify the denominator.

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Step 2.2.1

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

$\frac{\ln (5)}{\ln (\frac{75}{3})} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 2.2.2

Divide $75$ by $3$ .

$\frac{\ln (5)}{\ln (25)} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 2.3

Rewrite $\ln (25)$ as $\ln (5^{2})$ .

$\frac{\ln (5)}{\ln (5^{2})} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 2.4

Expand $\ln (5^{2})$ by moving $2$ outside the logarithm.

$\frac{\ln (5)}{2 \ln (5)} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 2.5

Cancel the common factor of $\ln (5)$ .

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Step 2.5.1

Cancel the common factor.

$\frac{\ln (5)}{2 \ln (5)} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 2.5.2

Rewrite the expression.

$\frac{1}{2} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

$\frac{1}{2} + \ln (9) - \frac{1}{2} - \ln (10) \log (9)$

Step 3

Combine fractions.

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Step 3.1

Combine the numerators over the common denominator.

$\ln (9) - \ln (10) \log (9) + \frac{1 - 1}{2}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Subtract $1$ from $1$ .

$\ln (9) - \ln (10) \log (9) + \frac{0}{2}$

Step 3.2.2

Divide $0$ by $2$ .

$\ln (9) - \ln (10) \log (9) + 0$

Step 3.2.3

Add $\ln (9)$ and $0$ .

$- \ln (10) \log (9) + \ln (9)$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \ln (10) \log (9) + \ln (9)$

Decimal Form:

$0$