Calculus Examples

$\ln (6) - \frac{3 e^{- \ln (6)} - 17 \ln (6)}{- 3 e^{- \ln (6)} - 17}$

Step 1

Simplify each term.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Simplify $- \ln (6)$ by moving $- 1$ inside the logarithm.

$\ln (6) - \frac{3 e^{\ln (6^{- 1})} - 17 \ln (6)}{- 3 e^{- \ln (6)} - 17}$

Step 1.1.2

Exponentiation and log are inverse functions.

$\ln (6) - \frac{3 \cdot 6^{- 1} - 17 \ln (6)}{- 3 e^{- \ln (6)} - 17}$

Step 1.1.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\ln (6) - \frac{3 \cdot \frac{1}{6} - 17 \ln (6)}{- 3 e^{- \ln (6)} - 17}$

Step 1.1.4

Cancel the common factor of $3$ .

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Step 1.1.4.1

Factor $3$ out of $6$ .

$\ln (6) - \frac{3 \cdot \frac{1}{3 (2)} - 17 \ln (6)}{- 3 e^{- \ln (6)} - 17}$

Step 1.1.4.2

Cancel the common factor.

$\ln (6) - \frac{3 \cdot \frac{1}{3 \cdot 2} - 17 \ln (6)}{- 3 e^{- \ln (6)} - 17}$

Step 1.1.4.3

Rewrite the expression.

$\ln (6) - \frac{\frac{1}{2} - 17 \ln (6)}{- 3 e^{- \ln (6)} - 17}$

Step 1.1.5

To write $- 17 \ln (6)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\ln (6) - \frac{\frac{1}{2} - 17 \ln (6) \cdot \frac{2}{2}}{- 3 e^{- \ln (6)} - 17}$

Step 1.1.6

Combine $- 17 \ln (6)$ and $\frac{2}{2}$ .

$\ln (6) - \frac{\frac{1}{2} + \frac{- 17 \ln (6) \cdot 2}{2}}{- 3 e^{- \ln (6)} - 17}$

Step 1.1.7

Combine the numerators over the common denominator.

$\ln (6) - \frac{\frac{1 - 17 \ln (6) \cdot 2}{2}}{- 3 e^{- \ln (6)} - 17}$

Step 1.1.8

Multiply $2$ by $- 17$ .

$\ln (6) - \frac{\frac{1 - 34 \ln (6)}{2}}{- 3 e^{- \ln (6)} - 17}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Simplify $- \ln (6)$ by moving $- 1$ inside the logarithm.

$\ln (6) - \frac{\frac{1 - 34 \ln (6)}{2}}{- 3 e^{\ln (6^{- 1})} - 17}$

Step 1.2.2

Exponentiation and log are inverse functions.

$\ln (6) - \frac{\frac{1 - 34 \ln (6)}{2}}{- 3 \cdot 6^{- 1} - 17}$

Step 1.2.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\ln (6) - \frac{\frac{1 - 34 \ln (6)}{2}}{- 3 \cdot \frac{1}{6} - 17}$

Step 1.2.4

Cancel the common factor of $3$ .

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Step 1.2.4.1

Factor $3$ out of $- 3$ .

$\ln (6) - \frac{\frac{1 - 34 \ln (6)}{2}}{3 (- 1) \cdot \frac{1}{6} - 17}$

Step 1.2.4.2

Factor $3$ out of $6$ .

$\ln (6) - \frac{\frac{1 - 34 \ln (6)}{2}}{3 \cdot - 1 \cdot \frac{1}{3 \cdot 2} - 17}$

Step 1.2.4.3

Cancel the common factor.

$\ln (6) - \frac{\frac{1 - 34 \ln (6)}{2}}{3 \cdot - 1 \cdot \frac{1}{3 \cdot 2} - 17}$

Step 1.2.4.4

Rewrite the expression.

$\ln (6) - \frac{\frac{1 - 34 \ln (6)}{2}}{- 1 \cdot \frac{1}{2} - 17}$

Step 1.2.5

Rewrite $- 1 (\frac{1}{2})$ as $- (\frac{1}{2})$ .

$\ln (6) - \frac{\frac{1 - 34 \ln (6)}{2}}{- (\frac{1}{2}) - 17}$

Step 1.2.6

To write $- 17$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\ln (6) - \frac{\frac{1 - 34 \ln (6)}{2}}{- \frac{1}{2} - 17 \cdot \frac{2}{2}}$

Step 1.2.7

Combine $- 17$ and $\frac{2}{2}$ .

$\ln (6) - \frac{\frac{1 - 34 \ln (6)}{2}}{- \frac{1}{2} + \frac{- 17 \cdot 2}{2}}$

Step 1.2.8

Combine the numerators over the common denominator.

$\ln (6) - \frac{\frac{1 - 34 \ln (6)}{2}}{\frac{- 1 - 17 \cdot 2}{2}}$

Step 1.2.9

Simplify the numerator.

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Step 1.2.9.1

Multiply $- 17$ by $2$ .

$\ln (6) - \frac{\frac{1 - 34 \ln (6)}{2}}{\frac{- 1 - 34}{2}}$

Step 1.2.9.2

Subtract $34$ from $- 1$ .

$\ln (6) - \frac{\frac{1 - 34 \ln (6)}{2}}{\frac{- 35}{2}}$

Step 1.2.10

Move the negative in front of the fraction.

$\ln (6) - \frac{\frac{1 - 34 \ln (6)}{2}}{- \frac{35}{2}}$

Step 1.3

Multiply the numerator by the reciprocal of the denominator.

$\ln (6) - (\frac{1 - 34 \ln (6)}{2} (- \frac{2}{35}))$

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Move the leading negative in $- \frac{2}{35}$ into the numerator.

$\ln (6) - (\frac{1 - 34 \ln (6)}{2} \cdot \frac{- 2}{35})$

Step 1.4.2

Factor $2$ out of $- 2$ .

$\ln (6) - (\frac{1 - 34 \ln (6)}{2} \cdot \frac{2 (- 1)}{35})$

Step 1.4.3

Cancel the common factor.

$\ln (6) - (\frac{1 - 34 \ln (6)}{2} \cdot \frac{2 \cdot - 1}{35})$

Step 1.4.4

Rewrite the expression.

$\ln (6) - ((1 - 34 \ln (6)) \frac{- 1}{35})$

Step 1.5

Move the negative in front of the fraction.

$\ln (6) - ((1 - 34 \ln (6)) (- \frac{1}{35}))$

Step 1.6

Apply the distributive property.

$\ln (6) - (1 (- \frac{1}{35}) - 34 \ln (6) (- \frac{1}{35}))$

Step 1.7

Multiply $- \frac{1}{35}$ by $1$ .

$\ln (6) - (- \frac{1}{35} - 34 \ln (6) (- \frac{1}{35}))$

Step 1.8

Multiply $- 34 \ln (6) (- \frac{1}{35})$ .

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Step 1.8.1

Multiply $- 1$ by $- 34$ .

$\ln (6) - (- \frac{1}{35} + 34 \ln (6) \frac{1}{35})$

Step 1.8.2

Simplify $34 \ln (6)$ by moving $34$ inside the logarithm.

$\ln (6) - (- \frac{1}{35} + \ln (6^{34}) \frac{1}{35})$

Step 1.8.3

Combine $\ln (6^{34})$ and $\frac{1}{35}$ .

$\ln (6) - (- \frac{1}{35} + \frac{\ln (6^{34})}{35})$

Step 1.9

Combine the numerators over the common denominator.

$\ln (6) - \frac{- 1 + \ln (6^{34})}{35}$

Step 1.10

Raise $6$ to the power of $34$ .

$\ln (6) - \frac{- 1 + \ln (286511799958070000000000000)}{35}$

Step 2

To write $\ln (6)$ as a fraction with a common denominator, multiply by $\frac{35}{35}$ .

$\ln (6) \cdot \frac{35}{35} - \frac{- 1 + \ln (286511799958070000000000000)}{35}$

Step 3

Combine fractions.

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Step 3.1

Combine $\ln (6)$ and $\frac{35}{35}$ .

$\frac{\ln (6) \cdot 35}{35} - \frac{- 1 + \ln (286511799958070000000000000)}{35}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{\ln (6) \cdot 35 - (- 1 + \ln (286511799958070000000000000))}{35}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $\ln (6) \cdot 35$ .

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Step 4.1.1

Reorder $\ln (6)$ and $35$ .

$\frac{35 \cdot \ln (6) - (- 1 + \ln (286511799958070000000000000))}{35}$

Step 4.1.2

Simplify $35 \cdot \ln (6)$ by moving $35$ inside the logarithm.

$\frac{\ln (6^{35}) - (- 1 + \ln (286511799958070000000000000))}{35}$

Step 4.2

Raise $6$ to the power of $35$ .

$\frac{\ln (1719070799748420000000000000) - (- 1 + \ln (286511799958070000000000000))}{35}$

Step 4.3

Apply the distributive property.

$\frac{\ln (1719070799748420000000000000) - - 1 - \ln (286511799958070000000000000)}{35}$

Step 4.4

Multiply $- 1$ by $- 1$ .

$\frac{\ln (1719070799748420000000000000) + 1 - \ln (286511799958070000000000000)}{35}$

Step 4.5

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

$\frac{\ln (\frac{1719070799748420000000000000}{286511799958070000000000000}) + 1}{35}$

Step 4.6

Divide $1719070799748420000000000000$ by $286511799958070000000000000$ .

$\frac{\ln (6) + 1}{35}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (6) + 1}{35}$

Decimal Form:

$0.07976455 \dots$