Calculus Examples

$lim_{x \to 8} \log_{13} (1 + 13^{- x})$

Step 1

Move the limit inside the logarithm.

$\log_{13} (lim_{x \to 8} 1 + 13^{- x})$

Step 2

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

$\log_{13} (lim_{x \to 8} 1 + lim_{x \to 8} 13^{- x})$

Step 3

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

$\log_{13} (1 + lim_{x \to 8} 13^{- x})$

Step 4

Move the limit into the exponent.

$\log_{13} (1 + 13^{lim_{x \to 8} - x})$

Step 5

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$\log_{13} (1 + 13^{- lim_{x \to 8} x})$

Step 6

Simplify terms.

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Step 6.1

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

$\log_{13} (1 + 13^{- 8})$

Step 6.2

Simplify the answer.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

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

$\log_{13} (1 + \frac{1}{13^{8}})$

Step 6.2.1.2

Raise $13$ to the power of $8$ .

$\log_{13} (1 + \frac{1}{815730721})$

Step 6.2.2

Write $1$ as a fraction with a common denominator.

$\log_{13} (\frac{815730721}{815730721} + \frac{1}{815730721})$

Step 6.2.3

Combine the numerators over the common denominator.

$\log_{13} (\frac{815730721 + 1}{815730721})$

Step 6.2.4

Add $815730721$ and $1$ .

$\log_{13} (\frac{815730722}{815730721})$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\log_{13} (\frac{815730722}{815730721})$

Decimal Form:

$4.779411 \cdot 10^{- 10}$