Calculus Examples

$15 (\ln (x) - \ln (x + 5)) - \frac{4}{15} \cdot (\ln (x) - \ln (x + 5))$

Step 1

Simplify each term.

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Step 1.1

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

$15 \ln (\frac{x}{x + 5}) - \frac{4}{15} \cdot (\ln (x) - \ln (x + 5))$

Step 1.2

Simplify $15 \ln (\frac{x}{x + 5})$ by moving $15$ inside the logarithm.

$\ln ({(\frac{x}{x + 5})}^{15}) - \frac{4}{15} \cdot (\ln (x) - \ln (x + 5))$

Step 1.3

Apply the product rule to $\frac{x}{x + 5}$ .

$\ln (\frac{x^{15}}{{(x + 5)}^{15}}) - \frac{4}{15} \cdot (\ln (x) - \ln (x + 5))$

Step 1.4

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

$\ln (\frac{x^{15}}{{(x + 5)}^{15}}) - \frac{4}{15} \cdot \ln (\frac{x}{x + 5})$

Step 1.5

Multiply $- \frac{4}{15} \ln (\frac{x}{x + 5})$ .

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Step 1.5.1

Reorder $\ln (\frac{x}{x + 5})$ and $\frac{4}{15}$ .

$\ln (\frac{x^{15}}{{(x + 5)}^{15}}) - (\frac{4}{15} \ln (\frac{x}{x + 5}))$

Step 1.5.2

Simplify $\frac{4}{15} \ln (\frac{x}{x + 5})$ by moving $\frac{4}{15}$ inside the logarithm.

$\ln (\frac{x^{15}}{{(x + 5)}^{15}}) - \ln ({(\frac{x}{x + 5})}^{\frac{4}{15}})$

Step 1.6

Apply the product rule to $\frac{x}{x + 5}$ .

$\ln (\frac{x^{15}}{{(x + 5)}^{15}}) - \ln (\frac{x^{\frac{4}{15}}}{{(x + 5)}^{\frac{4}{15}}})$

Step 2

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

$\ln (\frac{\frac{x^{15}}{{(x + 5)}^{15}}}{\frac{x^{\frac{4}{15}}}{{(x + 5)}^{\frac{4}{15}}}})$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{x^{15}}{{(x + 5)}^{15}} \cdot \frac{{(x + 5)}^{\frac{4}{15}}}{x^{\frac{4}{15}}})$

Step 4

Combine.

$\ln (\frac{x^{15} {(x + 5)}^{\frac{4}{15}}}{{(x + 5)}^{15} x^{\frac{4}{15}}})$

Step 5

Move ${(x + 5)}^{\frac{4}{15}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\ln (\frac{x^{15}}{{(x + 5)}^{15} x^{\frac{4}{15}} {(x + 5)}^{- \frac{4}{15}}})$

Step 6

Multiply ${(x + 5)}^{15}$ by ${(x + 5)}^{- \frac{4}{15}}$ by adding the exponents.

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Step 6.1

Move ${(x + 5)}^{- \frac{4}{15}}$ .

$\ln (\frac{x^{15}}{{(x + 5)}^{- \frac{4}{15}} {(x + 5)}^{15} x^{\frac{4}{15}}})$

Step 6.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\ln (\frac{x^{15}}{{(x + 5)}^{- \frac{4}{15} + 15} x^{\frac{4}{15}}})$

Step 6.3

To write $15$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$\ln (\frac{x^{15}}{{(x + 5)}^{- \frac{4}{15} + 15 \cdot \frac{15}{15}} x^{\frac{4}{15}}})$

Step 6.4

Combine $15$ and $\frac{15}{15}$ .

$\ln (\frac{x^{15}}{{(x + 5)}^{- \frac{4}{15} + \frac{15 \cdot 15}{15}} x^{\frac{4}{15}}})$

Step 6.5

Combine the numerators over the common denominator.

$\ln (\frac{x^{15}}{{(x + 5)}^{\frac{- 4 + 15 \cdot 15}{15}} x^{\frac{4}{15}}})$

Step 6.6

Simplify the numerator.

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Step 6.6.1

Multiply $15$ by $15$ .

$\ln (\frac{x^{15}}{{(x + 5)}^{\frac{- 4 + 225}{15}} x^{\frac{4}{15}}})$

Step 6.6.2

Add $- 4$ and $225$ .

$\ln (\frac{x^{15}}{{(x + 5)}^{\frac{221}{15}} x^{\frac{4}{15}}})$

Step 7

Move $x^{\frac{4}{15}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\ln (\frac{x^{15} x^{- \frac{4}{15}}}{{(x + 5)}^{\frac{221}{15}}})$

Step 8

Multiply $x^{15}$ by $x^{- \frac{4}{15}}$ by adding the exponents.

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Step 8.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\ln (\frac{x^{15 - \frac{4}{15}}}{{(x + 5)}^{\frac{221}{15}}})$

Step 8.2

To write $15$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$\ln (\frac{x^{15 \cdot \frac{15}{15} - \frac{4}{15}}}{{(x + 5)}^{\frac{221}{15}}})$

Step 8.3

Combine $15$ and $\frac{15}{15}$ .

$\ln (\frac{x^{\frac{15 \cdot 15}{15} - \frac{4}{15}}}{{(x + 5)}^{\frac{221}{15}}})$

Step 8.4

Combine the numerators over the common denominator.

$\ln (\frac{x^{\frac{15 \cdot 15 - 4}{15}}}{{(x + 5)}^{\frac{221}{15}}})$

Step 8.5

Simplify the numerator.

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Step 8.5.1

Multiply $15$ by $15$ .

$\ln (\frac{x^{\frac{225 - 4}{15}}}{{(x + 5)}^{\frac{221}{15}}})$

Step 8.5.2

Subtract $4$ from $225$ .

$\ln (\frac{x^{\frac{221}{15}}}{{(x + 5)}^{\frac{221}{15}}})$