Precalculus Examples

$\frac{2 \log (12)}{12} - \frac{\log (100)}{2}$

Step 1

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

$\frac{2 \log (2^{2} \cdot 3)}{12} - \frac{\log (100)}{2}$

Step 2

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

$\frac{2 (\log (2^{2}) + \log (3))}{12} - \frac{\log (100)}{2}$

Step 3

Cancel the common factors.

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Step 3.1

Factor $2$ out of $12$ .

$\frac{2 (\log (2^{2}) + \log (3))}{2 \cdot 6} - \frac{\log (100)}{2}$

Step 3.2

Cancel the common factor.

$\frac{2 (\log (2^{2}) + \log (3))}{2 \cdot 6} - \frac{\log (100)}{2}$

Step 3.3

Rewrite the expression.

$\frac{\log (2^{2}) + \log (3)}{6} - \frac{\log (100)}{2}$

Step 4

Simplify the numerator.

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Step 4.1

Raise $2$ to the power of $2$ .

$\frac{\log (4) + \log (3)}{6} - \frac{\log (100)}{2}$

Step 4.2

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

$\frac{\log (4 \cdot 3)}{6} - \frac{\log (100)}{2}$

Step 4.3

Multiply $4$ by $3$ .

$\frac{\log (12)}{6} - \frac{\log (100)}{2}$

Step 5

Rewrite $\frac{\log (12)}{6}$ as $\frac{1}{6} \log (12)$ .

$\frac{1}{6} \log (12) - \frac{\log (100)}{2}$

Step 6

Simplify $\frac{1}{6} \log (12)$ by moving $\frac{1}{6}$ inside the logarithm.

$\log (12^{\frac{1}{6}}) - \frac{\log (100)}{2}$

Step 7

Rewrite $\frac{\log (100)}{2}$ as $\frac{1}{2} \log (100)$ .

$\log (12^{\frac{1}{6}}) - (\frac{1}{2} \log (100))$

Step 8

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

$\log (12^{\frac{1}{6}}) - \log (100^{\frac{1}{2}})$

Step 9

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

$\log (12^{\frac{1}{6}}) - \log ({(10^{2})}^{\frac{1}{2}})$

Step 10

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

$\log (12^{\frac{1}{6}}) - \log (10^{2 (\frac{1}{2})})$

Step 11

Cancel the common factor of $2$ .

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Step 11.1

Cancel the common factor.

$\log (12^{\frac{1}{6}}) - \log (10^{2 (\frac{1}{2})})$

Step 11.2

Rewrite the expression.

$\log (12^{\frac{1}{6}}) - \log (10^{1})$

Step 12

Evaluate the exponent.

$\log (12^{\frac{1}{6}}) - \log (10)$

Step 13

Logarithm base $10$ of $10$ is $1$ .

$\log (12^{\frac{1}{6}}) - 1 \cdot 1$

Step 14

Multiply $- 1$ by $1$ .

$\log (12^{\frac{1}{6}}) - 1$

Step 15

The result can be shown in multiple forms.

Exact Form:

$\log (12^{\frac{1}{6}}) - 1$

Decimal Form:

$- 0.82013645 \dots$