Trigonometry Examples

$\log_{2} (\frac{4}{3}) + 2 \log_{2} (\sqrt{10})$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log_{2} (\sqrt{10})$ by moving $2$ inside the logarithm.

$\log_{2} (\frac{4}{3}) + \log_{2} ({\sqrt{10}}^{2})$

Step 1.2

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\log_{2} (\frac{4}{3}) + \log_{2} ({(10^{\frac{1}{2}})}^{2})$

Step 1.2.2

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

$\log_{2} (\frac{4}{3}) + \log_{2} (10^{\frac{1}{2} \cdot 2})$

Step 1.2.3

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

$\log_{2} (\frac{4}{3}) + \log_{2} (10^{\frac{2}{2}})$

Step 1.2.4

Cancel the common factor of $2$ .

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Step 1.2.4.1

Cancel the common factor.

$\log_{2} (\frac{4}{3}) + \log_{2} (10^{\frac{2}{2}})$

Step 1.2.4.2

Rewrite the expression.

$\log_{2} (\frac{4}{3}) + \log_{2} (10^{1})$

Step 1.2.5

Evaluate the exponent.

$\log_{2} (\frac{4}{3}) + \log_{2} (10)$

Step 2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{2} (\frac{4}{3} \cdot 10)$

Step 3

Multiply $\frac{4}{3} \cdot 10$ .

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Step 3.1

Combine $\frac{4}{3}$ and $10$ .

$\log_{2} (\frac{4 \cdot 10}{3})$

Step 3.2

Multiply $4$ by $10$ .

$\log_{2} (\frac{40}{3})$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\log_{2} (\frac{40}{3})$

Decimal Form:

$3.73696559 \dots$