Precalculus Examples

$0.5 + \log (9) + \frac{\ln (15) - \ln (3)}{\ln (75) - \ln (3)} - \frac{\ln (9)}{\ln (10)}$

Step 1

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

$0.5 + \log (9) + \frac{\ln (\frac{15}{3})}{\ln (75) - \ln (3)} - \frac{\ln (9)}{\ln (10)}$

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 $15$ .

$0.5 + \log (9) + \frac{\ln (\frac{3 \cdot 5}{3})}{\ln (75) - \ln (3)} - \frac{\ln (9)}{\ln (10)}$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $3$ out of $3$ .

$0.5 + \log (9) + \frac{\ln (\frac{3 \cdot 5}{3 (1)})}{\ln (75) - \ln (3)} - \frac{\ln (9)}{\ln (10)}$

Step 2.1.2.2

Cancel the common factor.

$0.5 + \log (9) + \frac{\ln (\frac{3 \cdot 5}{3 \cdot 1})}{\ln (75) - \ln (3)} - \frac{\ln (9)}{\ln (10)}$

Step 2.1.2.3

Rewrite the expression.

$0.5 + \log (9) + \frac{\ln (\frac{5}{1})}{\ln (75) - \ln (3)} - \frac{\ln (9)}{\ln (10)}$

Step 2.1.2.4

Divide $5$ by $1$ .

$0.5 + \log (9) + \frac{\ln (5)}{\ln (75) - \ln (3)} - \frac{\ln (9)}{\ln (10)}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

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

$0.5 + \log (9) + \frac{\ln (5)}{\ln (\frac{75}{3})} - \frac{\ln (9)}{\ln (10)}$

Step 2.2.2

Divide $75$ by $3$ .

$0.5 + \log (9) + \frac{\ln (5)}{\ln (25)} - \frac{\ln (9)}{\ln (10)}$

Step 2.3

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

$0.5 + \log (9) + \frac{\ln (5)}{\ln (5^{2})} - \frac{\ln (9)}{\ln (10)}$

Step 2.4

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

$0.5 + \log (9) + \frac{\ln (5)}{2 \ln (5)} - \frac{\ln (9)}{\ln (10)}$

Step 2.5

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

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Step 2.5.1

Cancel the common factor.

$0.5 + \log (9) + \frac{\ln (5)}{2 \ln (5)} - \frac{\ln (9)}{\ln (10)}$

Step 2.5.2

Rewrite the expression.

$0.5 + \log (9) + \frac{1}{2} - \frac{\ln (9)}{\ln (10)}$

Step 3

To write $0.5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$0.5 \cdot \frac{2}{2} + \frac{1}{2} + \log (9) - \frac{\ln (9)}{\ln (10)}$

Step 4

Combine $0.5$ and $\frac{2}{2}$ .

$\frac{0.5 \cdot 2}{2} + \frac{1}{2} + \log (9) - \frac{\ln (9)}{\ln (10)}$

Step 5

Combine the numerators over the common denominator.

$\frac{0.5 \cdot 2 + 1}{2} + \log (9) - \frac{\ln (9)}{\ln (10)}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $0.5$ by $2$ .

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

Step 6.2

Add $1$ and $1$ .

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

Step 7

Divide $2$ by $2$ .

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

Step 8

Find the common denominator.

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Step 8.1

Write $1$ as a fraction with denominator $1$ .

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

Step 8.2

Multiply $\frac{1}{1}$ by $\frac{\ln (10)}{\ln (10)}$ .

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

Step 8.3

Multiply $\frac{1}{1}$ by $\frac{\ln (10)}{\ln (10)}$ .

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

Step 8.4

Write $\log (9)$ as a fraction with denominator $1$ .

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

Step 8.5

Multiply $\frac{\log (9)}{1}$ by $\frac{\ln (10)}{\ln (10)}$ .

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

Step 8.6

Multiply $\frac{\log (9)}{1}$ by $\frac{\ln (10)}{\ln (10)}$ .

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

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

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

Step 11

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1$