Finite Math Examples

$\ln (\frac{7}{9}) + \ln (\frac{26}{28}) + \ln (\frac{63}{65}) + \ln (\frac{124}{125}) = m$

Step 1

Rewrite the equation as $m = \ln (\frac{7}{9}) + \ln (\frac{26}{28}) + \ln (\frac{63}{65}) + \ln (\frac{124}{125})$ .

$m = \ln (\frac{7}{9}) + \ln (\frac{26}{28}) + \ln (\frac{63}{65}) + \ln (\frac{124}{125})$

Step 2

Simplify $\ln (\frac{7}{9}) + \ln (\frac{26}{28}) + \ln (\frac{63}{65}) + \ln (\frac{124}{125})$ .

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Step 2.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$m = \ln (\frac{7}{9} \cdot \frac{26}{28}) + \ln (\frac{63}{65}) + \ln (\frac{124}{125})$

Step 2.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$m = \ln (\frac{7}{9} \cdot \frac{26}{28} \frac{63}{65}) + \ln (\frac{124}{125})$

Step 2.3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$m = \ln (\frac{7}{9} \cdot \frac{26}{28} \frac{63}{65} \frac{124}{125})$

Step 2.4

Cancel the common factor of $7$ .

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Step 2.4.1

Factor $7$ out of $28$ .

$m = \ln (\frac{7}{9} \cdot \frac{26}{7 (4)} \frac{63}{65} \frac{124}{125})$

Step 2.4.2

Cancel the common factor.

$m = \ln (\frac{7}{9} \cdot \frac{26}{7 \cdot 4} \frac{63}{65} \frac{124}{125})$

Step 2.4.3

Rewrite the expression.

$m = \ln (\frac{1}{9} \cdot \frac{26}{4} \frac{63}{65} \frac{124}{125})$

Step 2.5

Multiply $\frac{1}{9}$ by $\frac{26}{4}$ .

$m = \ln (\frac{26}{9 \cdot 4} \cdot \frac{63}{65} \frac{124}{125})$

Step 2.6

Multiply $9$ by $4$ .

$m = \ln (\frac{26}{36} \cdot \frac{63}{65} \frac{124}{125})$

Step 2.7

Cancel the common factor of $13$ .

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Step 2.7.1

Factor $13$ out of $26$ .

$m = \ln (\frac{13 (2)}{36} \cdot \frac{63}{65} \frac{124}{125})$

Step 2.7.2

Factor $13$ out of $65$ .

$m = \ln (\frac{13 \cdot 2}{36} \cdot \frac{63}{13 \cdot 5} \frac{124}{125})$

Step 2.7.3

Cancel the common factor.

$m = \ln (\frac{13 \cdot 2}{36} \cdot \frac{63}{13 \cdot 5} \frac{124}{125})$

Step 2.7.4

Rewrite the expression.

$m = \ln (\frac{2}{36} \cdot \frac{63}{5} \frac{124}{125})$

Step 2.8

Cancel the common factor of $9$ .

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Step 2.8.1

Factor $9$ out of $36$ .

$m = \ln (\frac{2}{9 (4)} \cdot \frac{63}{5} \frac{124}{125})$

Step 2.8.2

Factor $9$ out of $63$ .

$m = \ln (\frac{2}{9 \cdot 4} \cdot \frac{9 \cdot 7}{5} \frac{124}{125})$

Step 2.8.3

Cancel the common factor.

$m = \ln (\frac{2}{9 \cdot 4} \cdot \frac{9 \cdot 7}{5} \frac{124}{125})$

Step 2.8.4

Rewrite the expression.

$m = \ln (\frac{2}{4} \cdot \frac{7}{5} \frac{124}{125})$

Step 2.9

Multiply $\frac{2}{4}$ by $\frac{7}{5}$ .

$m = \ln (\frac{2 \cdot 7}{4 \cdot 5} \cdot \frac{124}{125})$

Step 2.10

Multiply.

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Step 2.10.1

Multiply $2$ by $7$ .

$m = \ln (\frac{14}{4 \cdot 5} \cdot \frac{124}{125})$

Step 2.10.2

Multiply $4$ by $5$ .

$m = \ln (\frac{14}{20} \cdot \frac{124}{125})$

Step 2.11

Cancel the common factor of $4$ .

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Step 2.11.1

Factor $4$ out of $20$ .

$m = \ln (\frac{14}{4 (5)} \cdot \frac{124}{125})$

Step 2.11.2

Factor $4$ out of $124$ .

$m = \ln (\frac{14}{4 \cdot 5} \cdot \frac{4 \cdot 31}{125})$

Step 2.11.3

Cancel the common factor.

$m = \ln (\frac{14}{4 \cdot 5} \cdot \frac{4 \cdot 31}{125})$

Step 2.11.4

Rewrite the expression.

$m = \ln (\frac{14}{5} \cdot \frac{31}{125})$

Step 2.12

Multiply $\frac{14}{5}$ by $\frac{31}{125}$ .

$m = \ln (\frac{14 \cdot 31}{5 \cdot 125})$

Step 2.13

Multiply.

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Step 2.13.1

Multiply $14$ by $31$ .

$m = \ln (\frac{434}{5 \cdot 125})$

Step 2.13.2

Multiply $5$ by $125$ .

$m = \ln (\frac{434}{625})$

Step 3

The result can be shown in multiple forms.

Exact Form:

$m = \ln (\frac{434}{625})$

Decimal Form:

$m = - 0.36470711 \dots$