Calculus Examples

$lim_{x \to 8} \ln ({(x + 1)}^{x}) - \ln ({(x)}^{x})$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$lim_{x \to 8} \ln ({(x + 1)}^{x}) - lim_{x \to 8} \ln ({(x)}^{x})$

Step 1.2

Move the limit inside the logarithm.

$\ln (lim_{x \to 8} {(x + 1)}^{x}) - lim_{x \to 8} \ln ({(x)}^{x})$

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

Rewrite ${(x + 1)}^{x}$ as $e^{\ln ({(x + 1)}^{x})}$ .

$\ln (lim_{x \to 8} e^{\ln ({(x + 1)}^{x})}) - lim_{x \to 8} \ln ({(x)}^{x})$

Step 2.2

Expand $\ln ({(x + 1)}^{x})$ by moving $x$ outside the logarithm.

$\ln (lim_{x \to 8} e^{x \ln (x + 1)}) - lim_{x \to 8} \ln ({(x)}^{x})$

Step 3

Evaluate the limit.

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Step 3.1

Move the limit into the exponent.

$\ln (e^{lim_{x \to 8} x \ln (x + 1)}) - lim_{x \to 8} \ln ({(x)}^{x})$

Step 3.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $8$ .

$\ln (e^{lim_{x \to 8} x \cdot lim_{x \to 8} \ln (x + 1)}) - lim_{x \to 8} \ln ({(x)}^{x})$

Step 3.3

Move the limit inside the logarithm.

$\ln (e^{lim_{x \to 8} x \cdot \ln (lim_{x \to 8} x + 1)}) - lim_{x \to 8} \ln ({(x)}^{x})$

Step 3.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\ln (e^{lim_{x \to 8} x \cdot \ln (lim_{x \to 8} x + lim_{x \to 8} 1)}) - lim_{x \to 8} \ln ({(x)}^{x})$

Step 3.5

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$\ln (e^{lim_{x \to 8} x \cdot \ln (lim_{x \to 8} x + 1)}) - lim_{x \to 8} \ln ({(x)}^{x})$

Step 3.6

Move the limit inside the logarithm.

$\ln (e^{lim_{x \to 8} x \cdot \ln (lim_{x \to 8} x + 1)}) - \ln (lim_{x \to 8} {(x)}^{x})$

Step 4

Use the properties of logarithms to simplify the limit.

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Step 4.1

Rewrite ${(x)}^{x}$ as $e^{\ln ({(x)}^{x})}$ .

$\ln (e^{lim_{x \to 8} x \cdot \ln (lim_{x \to 8} x + 1)}) - \ln (lim_{x \to 8} e^{\ln ({(x)}^{x})})$

Step 4.2

Expand $\ln ({(x)}^{x})$ by moving $x$ outside the logarithm.

$\ln (e^{lim_{x \to 8} x \cdot \ln (lim_{x \to 8} x + 1)}) - \ln (lim_{x \to 8} e^{x \ln (x)})$

Step 5

Evaluate the limit.

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Step 5.1

Move the limit into the exponent.

$\ln (e^{lim_{x \to 8} x \cdot \ln (lim_{x \to 8} x + 1)}) - \ln (e^{lim_{x \to 8} x \ln (x)})$

Step 5.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $8$ .

$\ln (e^{lim_{x \to 8} x \cdot \ln (lim_{x \to 8} x + 1)}) - \ln (e^{lim_{x \to 8} x \cdot lim_{x \to 8} \ln (x)})$

Step 5.3

Move the limit inside the logarithm.

$\ln (e^{lim_{x \to 8} x \cdot \ln (lim_{x \to 8} x + 1)}) - \ln (e^{lim_{x \to 8} x \cdot \ln (lim_{x \to 8} x)})$

Step 6

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\ln (e^{8 \cdot \ln (lim_{x \to 8} x + 1)}) - \ln (e^{lim_{x \to 8} x \cdot \ln (lim_{x \to 8} x)})$

Step 6.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\ln (e^{8 \cdot \ln (8 + 1)}) - \ln (e^{lim_{x \to 8} x \cdot \ln (lim_{x \to 8} x)})$

Step 6.3

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\ln (e^{8 \cdot \ln (8 + 1)}) - \ln (e^{8 \cdot \ln (lim_{x \to 8} x)})$

Step 6.4

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\ln (e^{8 \cdot \ln (8 + 1)}) - \ln (e^{8 \cdot \ln (8)})$

Step 7

Simplify the answer.

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Step 7.1

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

$\ln (\frac{e^{8 \cdot \ln (8 + 1)}}{e^{8 \cdot \ln (8)}})$

Step 7.2

Move $e^{8 \cdot \ln (8)}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\ln (e^{8 \cdot \ln (8 + 1)} e^{- 8 \cdot \ln (8)})$

Step 7.3

Multiply $e^{8 \cdot \ln (8 + 1)}$ by $e^{- 8 \cdot \ln (8)}$ by adding the exponents.

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Step 7.3.1

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

$\ln (e^{8 \cdot \ln (8 + 1) - 8 \cdot \ln (8)})$

Step 7.3.2

Add $8$ and $1$ .

$\ln (e^{8 \cdot \ln (9) - 8 \cdot \ln (8)})$

Step 7.3.3

Simplify each term.

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Step 7.3.3.1

Simplify $8 \ln (9)$ by moving $8$ inside the logarithm.

$\ln (e^{\ln (9^{8}) - 8 \cdot \ln (8)})$

Step 7.3.3.2

Raise $9$ to the power of $8$ .

$\ln (e^{\ln (43046721) - 8 \cdot \ln (8)})$

Step 7.3.3.3

Simplify $- 8 \ln (8)$ by moving $8$ inside the logarithm.

$\ln (e^{\ln (43046721) - \ln (8^{8})})$

Step 7.3.3.4

Raise $8$ to the power of $8$ .

$\ln (e^{\ln (43046721) - \ln (16777216)})$

Step 7.3.4

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

$\ln (e^{\ln (\frac{43046721}{16777216})})$

Step 7.4

Use logarithm rules to move $\ln (\frac{43046721}{16777216})$ out of the exponent.

$\ln (\frac{43046721}{16777216}) \ln (e)$

Step 7.5

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

$\ln (\frac{43046721}{16777216}) \cdot 1$

Step 7.6

Multiply $\ln (\frac{43046721}{16777216})$ by $1$ .

$\ln (\frac{43046721}{16777216})$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\ln (\frac{43046721}{16777216})$

Decimal Form:

$0.94226428 \dots$