Precalculus Examples

$\log (7^{- \frac{1}{5}}) \cdot 100$

Step 1

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

$\log (\frac{1}{7^{\frac{1}{5}}}) \cdot 100$

Step 2

Multiply $\log (\frac{1}{7^{\frac{1}{5}}}) \cdot 100$ .

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Step 2.1

Reorder $\log (\frac{1}{7^{\frac{1}{5}}})$ and $100$ .

$100 \cdot \log (\frac{1}{7^{\frac{1}{5}}})$

Step 2.2

Simplify $100 \cdot \log (\frac{1}{7^{\frac{1}{5}}})$ by moving $100$ inside the logarithm.

$\log ({(\frac{1}{7^{\frac{1}{5}}})}^{100})$

Step 3

Simplify the expression.

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Step 3.1

Apply the product rule to $\frac{1}{7^{\frac{1}{5}}}$ .

$\log (\frac{1^{100}}{{(7^{\frac{1}{5}})}^{100}})$

Step 3.2

One to any power is one.

$\log (\frac{1}{{(7^{\frac{1}{5}})}^{100}})$

Step 4

Simplify the denominator.

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Step 4.1

Multiply the exponents in ${(7^{\frac{1}{5}})}^{100}$ .

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Step 4.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log (\frac{1}{7^{\frac{1}{5} \cdot 100}})$

Step 4.1.2

Cancel the common factor of $5$ .

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Step 4.1.2.1

Factor $5$ out of $100$ .

$\log (\frac{1}{7^{\frac{1}{5} \cdot (5 (20))}})$

Step 4.1.2.2

Cancel the common factor.

$\log (\frac{1}{7^{\frac{1}{5} \cdot (5 \cdot 20)}})$

Step 4.1.2.3

Rewrite the expression.

$\log (\frac{1}{7^{20}})$

Step 4.2

Raise $7$ to the power of $20$ .

$\log (\frac{1}{79792266297612000})$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\log (\frac{1}{79792266297612000})$

Decimal Form:

$- 16.90196080 \dots$