Precalculus Examples

$\log_{2} (80) \log_{2} (10)$

Step 1

Rewrite $\log_{2} (80)$ as $\log_{2} (2^{4} \cdot 5)$ .

$\log_{2} (2^{4} \cdot 5) \log_{2} (10)$

Step 2

Rewrite $\log_{2} (2^{4} \cdot 5)$ as $\log_{2} (2^{4}) + \log_{2} (5)$ .

$(\log_{2} (2^{4}) + \log_{2} (5)) \log_{2} (10)$

Step 3

Use logarithm rules to move $4$ out of the exponent.

$(4 \log_{2} (2) + \log_{2} (5)) \log_{2} (10)$

Step 4

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

$(4 \cdot 1 + \log_{2} (5)) \log_{2} (10)$

Step 5

Multiply $4$ by $1$ .

$(4 + \log_{2} (5)) \log_{2} (10)$

Step 6

Rewrite $\log_{2} (10)$ as $\log_{2} (2 (5))$ .

$(4 + \log_{2} (5)) \log_{2} (2 (5))$

Step 7

Rewrite $\log_{2} (2 (5))$ as $\log_{2} (2) + \log_{2} (5)$ .

$(4 + \log_{2} (5)) (\log_{2} (2) + \log_{2} (5))$

Step 8

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

$(4 + \log_{2} (5)) (1 + \log_{2} (5))$

Step 9

Expand $(4 + \log_{2} (5)) (1 + \log_{2} (5))$ using the FOIL Method.

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Step 9.1

Apply the distributive property.

$4 (1 + \log_{2} (5)) + \log_{2} (5) (1 + \log_{2} (5))$

Step 9.2

Apply the distributive property.

$4 \cdot 1 + 4 \log_{2} (5) + \log_{2} (5) (1 + \log_{2} (5))$

Step 9.3

Apply the distributive property.

$4 \cdot 1 + 4 \log_{2} (5) + \log_{2} (5) \cdot 1 + \log_{2} (5) \log_{2} (5)$

Step 10

Simplify and combine like terms.

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Step 10.1

Simplify each term.

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Step 10.1.1

Multiply $4$ by $1$ .

$4 + 4 \log_{2} (5) + \log_{2} (5) \cdot 1 + \log_{2} (5) \log_{2} (5)$

Step 10.1.2

Multiply $\log_{2} (5)$ by $1$ .

$4 + 4 \log_{2} (5) + \log_{2} (5) + \log_{2} (5) \log_{2} (5)$

Step 10.1.3

Multiply $\log_{2} (5) \log_{2} (5)$ .

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Step 10.1.3.1

Raise $\log_{2} (5)$ to the power of $1$ .

$4 + 4 \log_{2} (5) + \log_{2} (5) + {\log_{2}}^{1} (5) \log_{2} (5)$

Step 10.1.3.2

Raise $\log_{2} (5)$ to the power of $1$ .

$4 + 4 \log_{2} (5) + \log_{2} (5) + {\log_{2}}^{1} (5) {\log_{2}}^{1} (5)$

Step 10.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 + 4 \log_{2} (5) + \log_{2} (5) + {\log_{2} (5)}^{1 + 1}$

Step 10.1.3.4

Add $1$ and $1$ .

$4 + 4 \log_{2} (5) + \log_{2} (5) + {\log_{2}}^{2} (5)$

Step 10.2

Add $4 \log_{2} (5)$ and $\log_{2} (5)$ .

$4 + 5 \log_{2} (5) + {\log_{2}}^{2} (5)$