Precalculus Examples

$\frac{1}{2} \cdot \log_{2} (100) - \frac{1}{2} \cdot \log_{2} (25)$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{2} \log_{2} (100)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{2} (100^{\frac{1}{2}}) - \frac{1}{2} \cdot \log_{2} (25)$

Step 1.2

Rewrite $100$ as $10^{2}$ .

$\log_{2} ({(10^{2})}^{\frac{1}{2}}) - \frac{1}{2} \cdot \log_{2} (25)$

Step 1.3

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

$\log_{2} (10^{2 (\frac{1}{2})}) - \frac{1}{2} \cdot \log_{2} (25)$

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Cancel the common factor.

$\log_{2} (10^{2 (\frac{1}{2})}) - \frac{1}{2} \cdot \log_{2} (25)$

Step 1.4.2

Rewrite the expression.

$\log_{2} (10^{1}) - \frac{1}{2} \cdot \log_{2} (25)$

Step 1.5

Evaluate the exponent.

$\log_{2} (10) - \frac{1}{2} \cdot \log_{2} (25)$

Step 1.6

Multiply $- \frac{1}{2} \log_{2} (25)$ .

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Step 1.6.1

Reorder $\log_{2} (25)$ and $\frac{1}{2}$ .

$\log_{2} (10) - (\frac{1}{2} \log_{2} (25))$

Step 1.6.2

Simplify $\frac{1}{2} \log_{2} (25)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{2} (10) - \log_{2} (25^{\frac{1}{2}})$

Step 1.7

Rewrite $25$ as $5^{2}$ .

$\log_{2} (10) - \log_{2} ({(5^{2})}^{\frac{1}{2}})$

Step 1.8

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

$\log_{2} (10) - \log_{2} (5^{2 (\frac{1}{2})})$

Step 1.9

Cancel the common factor of $2$ .

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Step 1.9.1

Cancel the common factor.

$\log_{2} (10) - \log_{2} (5^{2 (\frac{1}{2})})$

Step 1.9.2

Rewrite the expression.

$\log_{2} (10) - \log_{2} (5^{1})$

Step 1.10

Evaluate the exponent.

$\log_{2} (10) - \log_{2} (5)$

Step 2

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

$\log_{2} (\frac{10}{5})$

Step 3

Logarithm base $2$ of $\frac{10}{5}$ is $1$ .

$1$