Calculus Examples

$\log_{5} (\frac{5 p}{12 k})$

Step 1

Rewrite $\log_{5} (\frac{5 p}{12 k})$ as $\log_{5} (5 p) - \log_{5} (12 k)$ .

$\log_{5} (5 p) - \log_{5} (12 k)$

Step 2

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

$\log_{5} (5) + \log_{5} (p) - \log_{5} (12 k)$

Step 3

Logarithm base $5$ of $5$ is $1$ .

$1 + \log_{5} (p) - \log_{5} (12 k)$

Step 4

Rewrite $\log_{5} (12 k)$ as $\log_{5} (12) + \log_{5} (k)$ .

$1 + \log_{5} (p) - (\log_{5} (12) + \log_{5} (k))$

Step 5

Simplify each term.

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Step 5.1

Simplify each term.

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Step 5.1.1

Rewrite $\log_{5} (12)$ as $\log_{5} (2^{2} \cdot 3)$ .

$1 + \log_{5} (p) - (\log_{5} (2^{2} \cdot 3) + \log_{5} (k))$

Step 5.1.2

Rewrite $\log_{5} (2^{2} \cdot 3)$ as $\log_{5} (2^{2}) + \log_{5} (3)$ .

$1 + \log_{5} (p) - (\log_{5} (2^{2}) + \log_{5} (3) + \log_{5} (k))$

Step 5.1.3

Expand $\log_{5} (2^{2})$ by moving $2$ outside the logarithm.

$1 + \log_{5} (p) - (2 \log_{5} (2) + \log_{5} (3) + \log_{5} (k))$

Step 5.2

Apply the distributive property.

$1 + \log_{5} (p) - (2 \log_{5} (2)) - \log_{5} (3) - \log_{5} (k)$

Step 5.3

Multiply $2$ by $- 1$ .

$1 + \log_{5} (p) - 2 \log_{5} (2) - \log_{5} (3) - \log_{5} (k)$