Precalculus Examples

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

Step 1

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

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

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 $30$ .

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

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $3$ out of $3$ .

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

Step 2.1.2.2

Cancel the common factor.

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

Step 2.1.2.3

Rewrite the expression.

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

Step 2.1.2.4

Divide $10$ by $1$ .

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

Step 2.2

Simplify the denominator.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

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

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

Step 2.2.1.2

Multiply $- 1$ by $0$ .

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

Step 2.2.2

Add $100$ and $0$ .

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

Step 2.3

Multiply $- \frac{1}{2} \ln (10)$ .

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Step 2.3.1

Reorder $\ln (10)$ and $\frac{1}{2}$ .

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

Step 2.3.2

Simplify $\frac{1}{2} \ln (10)$ by moving $\frac{1}{2}$ inside the logarithm.

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

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

To write $- \ln (10^{\frac{1}{2}}) \log (9)$ as a fraction with a common denominator, multiply by $\frac{\ln (100)}{\ln (100)}$ .

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

Step 6

Combine $- \ln (10^{\frac{1}{2}}) \log (9)$ and $\frac{\ln (100)}{\ln (100)}$ .

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.59861228 \dots$