Precalculus Examples

$\frac{\ln (30) - \ln (3)}{\ln (100) - \ln (1)} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 1

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

$\frac{\ln (\frac{30}{3})}{\ln (100) - \ln (1)} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 2

Simplify each term.

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Step 2.1

Cancel the common factor of $30$ and $3$ .

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Step 2.1.1

Factor $3$ out of $30$ .

$\frac{\ln (\frac{3 \cdot 10}{3})}{\ln (100) - \ln (1)} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $3$ out of $3$ .

$\frac{\ln (\frac{3 \cdot 10}{3 (1)})}{\ln (100) - \ln (1)} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 2.1.2.2

Cancel the common factor.

$\frac{\ln (\frac{3 \cdot 10}{3 \cdot 1})}{\ln (100) - \ln (1)} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 2.1.2.3

Rewrite the expression.

$\frac{\ln (\frac{10}{1})}{\ln (100) - \ln (1)} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 2.1.2.4

Divide $10$ by $1$ .

$\frac{\ln (10)}{\ln (100) - \ln (1)} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 2.2

Simplify the denominator.

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Step 2.2.1

The natural logarithm of $1$ is $0$ .

$\frac{\ln (10)}{\ln (100) - 0} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 2.2.2

Multiply $- 1$ by $0$ .

$\frac{\ln (10)}{\ln (100) + 0} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

Step 2.2.3

Add $\ln (100)$ and $0$ .

$\frac{\ln (10)}{\ln (100)} + \ln (9) - \frac{1}{2} - \ln (10) \cdot \log (9)$

$\frac{\ln (10)}{\ln (100)} + \ln (9) - \frac{1}{2} - \ln (10) \log (9)$

Step 3

To write $\ln (9)$ as a fraction with a common denominator, multiply by $\frac{\ln (100)}{\ln (100)}$ .

$\frac{\ln (10)}{\ln (100)} + \frac{\ln (9) \ln (100)}{\ln (100)} - \frac{1}{2} - \ln (10) \log (9)$

Step 4

Combine the numerators over the common denominator.

$\frac{\ln (10) + \ln (9) \ln (100)}{\ln (100)} - \frac{1}{2} - \ln (10) \log (9)$

Step 5

To write $\frac{\ln (10) + \ln (9) \ln (100)}{\ln (100)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\ln (10) + \ln (9) \ln (100)}{\ln (100)} \cdot \frac{2}{2} - \frac{1}{2} - \ln (10) \log (9)$

Step 6

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{\ln (100)}{\ln (100)}$ .

$\frac{\ln (10) + \ln (9) \ln (100)}{\ln (100)} \cdot \frac{2}{2} - \frac{1}{2} \cdot \frac{\ln (100)}{\ln (100)} - \ln (10) \log (9)$

Step 7

Write each expression with a common denominator of $\ln (100) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

Multiply $\frac{\ln (10) + \ln (9) \ln (100)}{\ln (100)}$ by $\frac{2}{2}$ .

$\frac{(\ln (10) + \ln (9) \ln (100)) \cdot 2}{\ln (100) \cdot 2} - \frac{1}{2} \cdot \frac{\ln (100)}{\ln (100)} - \ln (10) \log (9)$

Step 7.2

Reorder $\ln (100)$ and $2$ .

$\frac{(\ln (10) + \ln (9) \ln (100)) \cdot 2}{2 \cdot \ln (100)} - \frac{1}{2} \cdot \frac{\ln (100)}{\ln (100)} - \ln (10) \log (9)$

Step 7.3

Simplify $2 \cdot \ln (100)$ by moving $2$ inside the logarithm.

$\frac{(\ln (10) + \ln (9) \ln (100)) \cdot 2}{\ln (100^{2})} - \frac{1}{2} \cdot \frac{\ln (100)}{\ln (100)} - \ln (10) \log (9)$

Step 7.4

Multiply $\frac{1}{2}$ by $\frac{\ln (100)}{\ln (100)}$ .

$\frac{(\ln (10) + \ln (9) \ln (100)) \cdot 2}{\ln (100^{2})} - \frac{\ln (100)}{2 \ln (100)} - \ln (10) \log (9)$

Step 7.5

Simplify $2 \ln (100)$ by moving $2$ inside the logarithm.

$\frac{(\ln (10) + \ln (9) \ln (100)) \cdot 2}{\ln (100^{2})} - \frac{\ln (100)}{\ln (100^{2})} - \ln (10) \log (9)$

Step 8

Combine the numerators over the common denominator.

$\frac{(\ln (10) + \ln (9) \ln (100)) \cdot 2 - \ln (100)}{\ln (100^{2})} - \ln (10) \log (9)$

Step 9

Simplify each term.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Apply the distributive property.

$\frac{\ln (10) \cdot 2 + \ln (9) \ln (100) \cdot 2 - \ln (100)}{\ln (100^{2})} - \ln (10) \log (9)$

Step 9.1.2

Multiply $\ln (10) \cdot 2$ .

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Step 9.1.2.1

Reorder $\ln (10)$ and $2$ .

$\frac{2 \cdot \ln (10) + \ln (9) \ln (100) \cdot 2 - \ln (100)}{\ln (100^{2})} - \ln (10) \log (9)$

Step 9.1.2.2

Simplify $2 \cdot \ln (10)$ by moving $2$ inside the logarithm.

$\frac{\ln (10^{2}) + \ln (9) \ln (100) \cdot 2 - \ln (100)}{\ln (100^{2})} - \ln (10) \log (9)$

Step 9.1.3

Simplify $2 \ln (9)$ by moving $2$ inside the logarithm.

$\frac{\ln (10^{2}) + \ln (9^{2}) \ln (100) - \ln (100)}{\ln (100^{2})} - \ln (10) \log (9)$

Step 9.1.4

Simplify each term.

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Step 9.1.4.1

Raise $10$ to the power of $2$ .

$\frac{\ln (100) + \ln (9^{2}) \ln (100) - \ln (100)}{\ln (100^{2})} - \ln (10) \log (9)$

Step 9.1.4.2

Raise $9$ to the power of $2$ .

$\frac{\ln (100) + \ln (81) \ln (100) - \ln (100)}{\ln (100^{2})} - \ln (10) \log (9)$

Step 9.1.5

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

$\frac{\ln (81) \ln (100) + \ln (\frac{100}{100})}{\ln (100^{2})} - \ln (10) \log (9)$

Step 9.1.6

Rewrite $\ln (81) \ln (100) + \ln (\frac{100}{100})$ in a factored form.

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Step 9.1.6.1

Divide $100$ by $100$ .

$\frac{\ln (81) \ln (100) + \ln (1)}{\ln (100^{2})} - \ln (10) \log (9)$

Step 9.1.6.2

The natural logarithm of $1$ is $0$ .

$\frac{\ln (81) \ln (100) + 0}{\ln (100^{2})} - \ln (10) \log (9)$

Step 9.1.6.3

Add $\ln (81) \ln (100)$ and $0$ .

$\frac{\ln (81) \ln (100)}{\ln (100^{2})} - \ln (10) \log (9)$

Step 9.2

Raise $100$ to the power of $2$ .

$\frac{\ln (81) \ln (100)}{\ln (10000)} - \ln (10) \log (9)$

Step 10

To write $- \ln (10) \log (9)$ as a fraction with a common denominator, multiply by $\frac{\ln (10000)}{\ln (10000)}$ .

$\frac{\ln (81) \ln (100)}{\ln (10000)} - \ln (10) \log (9) \cdot \frac{\ln (10000)}{\ln (10000)}$

Step 11

Combine $- \ln (10) \log (9)$ and $\frac{\ln (10000)}{\ln (10000)}$ .

$\frac{\ln (81) \ln (100)}{\ln (10000)} + \frac{- \ln (10) \log (9) \ln (10000)}{\ln (10000)}$

Step 12

Combine the numerators over the common denominator.

$\frac{\ln (81) \ln (100) - \ln (10) \log (9) \ln (10000)}{\ln (10000)}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (81) \ln (100) - \ln (10) \log (9) \ln (10000)}{\ln (10000)}$

Decimal Form:

$0$