Trigonometry Examples

$3 \ln (2) + \frac{1}{2} \cdot \ln (49) - \ln (14)$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \ln (2)$ by moving $3$ inside the logarithm.

$\ln (2^{3}) + \frac{1}{2} \cdot \ln (49) - \ln (14)$

Step 1.2

Raise $2$ to the power of $3$ .

$\ln (8) + \frac{1}{2} \cdot \ln (49) - \ln (14)$

Step 1.3

Simplify $\frac{1}{2} \ln (49)$ by moving $\frac{1}{2}$ inside the logarithm.

$\ln (8) + \ln (49^{\frac{1}{2}}) - \ln (14)$

Step 1.4

Rewrite $49$ as $7^{2}$ .

$\ln (8) + \ln ({(7^{2})}^{\frac{1}{2}}) - \ln (14)$

Step 1.5

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

$\ln (8) + \ln (7^{2 (\frac{1}{2})}) - \ln (14)$

Step 1.6

Cancel the common factor of $2$ .

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Step 1.6.1

Cancel the common factor.

$\ln (8) + \ln (7^{2 (\frac{1}{2})}) - \ln (14)$

Step 1.6.2

Rewrite the expression.

$\ln (8) + \ln (7^{1}) - \ln (14)$

Step 1.7

Evaluate the exponent.

$\ln (8) + \ln (7) - \ln (14)$

Step 2

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

$\ln (8 \cdot 7) - \ln (14)$

Step 3

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

$\ln (\frac{8 \cdot 7}{14})$

Step 4

Cancel the common factor of $8$ and $14$ .

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Step 4.1

Factor $2$ out of $8 \cdot 7$ .

$\ln (\frac{2 (4 \cdot 7)}{14})$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $2$ out of $14$ .

$\ln (\frac{2 (4 \cdot 7)}{2 (7)})$

Step 4.2.2

Cancel the common factor.

$\ln (\frac{2 (4 \cdot 7)}{2 \cdot 7})$

Step 4.2.3

Rewrite the expression.

$\ln (\frac{4 \cdot 7}{7})$

Step 5

Cancel the common factor of $7$ .

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Step 5.1

Cancel the common factor.

$\ln (\frac{4 \cdot 7}{7})$

Step 5.2

Divide $4$ by $1$ .

$\ln (4)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\ln (4)$

Decimal Form:

$1.38629436 \dots$