Precalculus Examples

$- \frac{5}{6} \cdot \log_{2} (2 m^{2}) + \frac{1}{2} \cdot \log_{2} (4 m^{2})$

Step 1

Multiply $- \frac{5}{6} \log_{2} (2 m^{2})$ .

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Step 1.1

Reorder $\log_{2} (2 m^{2})$ and $\frac{5}{6}$ .

$- (\frac{5}{6} \log_{2} (2 m^{2})) + \frac{1}{2} \cdot \log_{2} (4 m^{2})$

Step 1.2

Simplify $\frac{5}{6} \log_{2} (2 m^{2})$ by moving $\frac{5}{6}$ inside the logarithm.

$- \log_{2} ({(2 m^{2})}^{\frac{5}{6}}) + \frac{1}{2} \cdot \log_{2} (4 m^{2})$

Step 2

Apply the product rule to $2 m^{2}$ .

$- \log_{2} (2^{\frac{5}{6}} {(m^{2})}^{\frac{5}{6}}) + \frac{1}{2} \cdot \log_{2} (4 m^{2})$

Step 3

Rewrite $\log_{2} (2^{\frac{5}{6}} {(m^{2})}^{\frac{5}{6}})$ as $\log_{2} (2^{\frac{5}{6}}) + \log_{2} ({(m^{2})}^{\frac{5}{6}})$ .

$- (\log_{2} (2^{\frac{5}{6}}) + \log_{2} ({(m^{2})}^{\frac{5}{6}})) + \frac{1}{2} \cdot \log_{2} (4 m^{2})$

Step 4

Use logarithm rules to move $\frac{5}{6}$ out of the exponent.

$- (\frac{5}{6} \log_{2} (2) + \log_{2} ({(m^{2})}^{\frac{5}{6}})) + \frac{1}{2} \cdot \log_{2} (4 m^{2})$

Step 5

Logarithm base $2$ of $2$ is $1$ .

$- (\frac{5}{6} \cdot 1 + \log_{2} ({(m^{2})}^{\frac{5}{6}})) + \frac{1}{2} \cdot \log_{2} (4 m^{2})$

Step 6

Multiply $\frac{5}{6}$ by $1$ .

$- (\frac{5}{6} + \log_{2} ({(m^{2})}^{\frac{5}{6}})) + \frac{1}{2} \cdot \log_{2} (4 m^{2})$

Step 7

Multiply the exponents in ${(m^{2})}^{\frac{5}{6}}$ .

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Step 7.1

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

$- (\frac{5}{6} + \log_{2} (m^{2 (\frac{5}{6})})) + \frac{1}{2} \cdot \log_{2} (4 m^{2})$

Step 7.2

Cancel the common factor of $2$ .

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Step 7.2.1

Factor $2$ out of $6$ .

$- (\frac{5}{6} + \log_{2} (m^{2 \frac{5}{2 (3)}})) + \frac{1}{2} \cdot \log_{2} (4 m^{2})$

Step 7.2.2

Cancel the common factor.

$- (\frac{5}{6} + \log_{2} (m^{2 \frac{5}{2 \cdot 3}})) + \frac{1}{2} \cdot \log_{2} (4 m^{2})$

Step 7.2.3

Rewrite the expression.

$- (\frac{5}{6} + \log_{2} (m^{\frac{5}{3}})) + \frac{1}{2} \cdot \log_{2} (4 m^{2})$

Step 8

Apply the distributive property.

$- \frac{5}{6} - \log_{2} (m^{\frac{5}{3}}) + \frac{1}{2} \cdot \log_{2} (4 m^{2})$

Step 9

Simplify $\frac{1}{2} \log_{2} (4 m^{2})$ by moving $\frac{1}{2}$ inside the logarithm.

$- \frac{5}{6} - \log_{2} (m^{\frac{5}{3}}) + \log_{2} ({(4 m^{2})}^{\frac{1}{2}})$

Step 10

Apply the product rule to $4 m^{2}$ .

$- \frac{5}{6} - \log_{2} (m^{\frac{5}{3}}) + \log_{2} (4^{\frac{1}{2}} {(m^{2})}^{\frac{1}{2}})$

Step 11

Rewrite $4$ as $2^{2}$ .

$- \frac{5}{6} - \log_{2} (m^{\frac{5}{3}}) + \log_{2} ({(2^{2})}^{\frac{1}{2}} {(m^{2})}^{\frac{1}{2}})$

Step 12

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

$- \frac{5}{6} - \log_{2} (m^{\frac{5}{3}}) + \log_{2} (2^{2 (\frac{1}{2})} {(m^{2})}^{\frac{1}{2}})$

Step 13

Cancel the common factor of $2$ .

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Step 13.1

Cancel the common factor.

$- \frac{5}{6} - \log_{2} (m^{\frac{5}{3}}) + \log_{2} (2^{2 (\frac{1}{2})} {(m^{2})}^{\frac{1}{2}})$

Step 13.2

Rewrite the expression.

$- \frac{5}{6} - \log_{2} (m^{\frac{5}{3}}) + \log_{2} (2^{1} {(m^{2})}^{\frac{1}{2}})$

Step 14

Evaluate the exponent.

$- \frac{5}{6} - \log_{2} (m^{\frac{5}{3}}) + \log_{2} (2 {(m^{2})}^{\frac{1}{2}})$

Step 15

Multiply the exponents in ${(m^{2})}^{\frac{1}{2}}$ .

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Step 15.1

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

$- \frac{5}{6} - \log_{2} (m^{\frac{5}{3}}) + \log_{2} (2 m^{2 (\frac{1}{2})})$

Step 15.2

Cancel the common factor of $2$ .

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Step 15.2.1

Cancel the common factor.

$- \frac{5}{6} - \log_{2} (m^{\frac{5}{3}}) + \log_{2} (2 m^{2 (\frac{1}{2})})$

Step 15.2.2

Rewrite the expression.

$- \frac{5}{6} - \log_{2} (m^{\frac{5}{3}}) + \log_{2} (2 m^{1})$

Step 16

Simplify.

$- \frac{5}{6} - \log_{2} (m^{\frac{5}{3}}) + \log_{2} (2 m)$