Precalculus Examples

$- 1 + 100 \log (\sqrt{2.09})$

Step 1

Simplify each term.

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Step 1.1

Simplify $100 \log (\sqrt{2.09})$ by moving $100$ inside the logarithm.

$- 1 + \log ({\sqrt{2.09}}^{100})$

Step 1.2

Rewrite ${\sqrt{2.09}}^{100}$ as ${2.09}^{50}$ .

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2.09}$ as ${2.09}^{\frac{1}{2}}$ .

$- 1 + \log ({({2.09}^{\frac{1}{2}})}^{100})$

Step 1.2.2

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

$- 1 + \log ({2.09}^{\frac{1}{2} \cdot 100})$

Step 1.2.3

Combine $\frac{1}{2}$ and $100$ .

$- 1 + \log ({2.09}^{\frac{100}{2}})$

Step 1.2.4

Cancel the common factor of $100$ and $2$ .

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Step 1.2.4.1

Factor $2$ out of $100$ .

$- 1 + \log ({2.09}^{\frac{2 \cdot 50}{2}})$

Step 1.2.4.2

Cancel the common factors.

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Step 1.2.4.2.1

Factor $2$ out of $2$ .

$- 1 + \log ({2.09}^{\frac{2 \cdot 50}{2 (1)}})$

Step 1.2.4.2.2

Cancel the common factor.

$- 1 + \log ({2.09}^{\frac{2 \cdot 50}{2 \cdot 1}})$

Step 1.2.4.2.3

Rewrite the expression.

$- 1 + \log ({2.09}^{\frac{50}{1}})$

Step 1.2.4.2.4

Divide $50$ by $1$ .

$- 1 + \log ({2.09}^{50})$

Step 1.3

Raise $2.09$ to the power of $50$ .

$- 1 + \log (10169844337796600)$

Step 1.4

Log base $10$ of $10169844337796600$ is approximately $16.0073143$ .

$- 1 + 16.0073143$

Step 2

Add $- 1$ and $16.0073143$ .

$15.0073143$