Precalculus Examples

$\frac{- \ln (7) + 2 \ln (3)}{2 \ln (7) + \ln (3)}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{- \ln (7) + \ln (3^{2})}{2 \ln (7) + \ln (3)}$

Step 1.2

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

$\frac{- \ln (7) + \ln (9)}{2 \ln (7) + \ln (3)}$

Step 2

Simplify the denominator.

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Step 2.1

Simplify $2 \ln (7)$ by moving $2$ inside the logarithm.

$\frac{- \ln (7) + \ln (9)}{\ln (7^{2}) + \ln (3)}$

Step 2.2

Raise $7$ to the power of $2$ .

$\frac{- \ln (7) + \ln (9)}{\ln (49) + \ln (3)}$

Step 2.3

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

$\frac{- \ln (7) + \ln (9)}{\ln (49 \cdot 3)}$

Step 2.4

Multiply $49$ by $3$ .

$\frac{- \ln (7) + \ln (9)}{\ln (147)}$

Step 3

Simplify with factoring out.

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Step 3.1

Factor $- 1$ out of $- \ln (7)$ .

$\frac{- (\ln (7)) + \ln (9)}{\ln (147)}$

Step 3.2

Factor $- 1$ out of $\ln (9)$ .

$\frac{- (\ln (7)) - 1 (- \ln (9))}{\ln (147)}$

Step 3.3

Factor $- 1$ out of $- (\ln (7)) - 1 (- \ln (9))$ .

$\frac{- (\ln (7) - \ln (9))}{\ln (147)}$

Step 3.4

Simplify the expression.

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Step 3.4.1

Rewrite $- (\ln (7) - \ln (9))$ as $- 1 (\ln (7) - \ln (9))$ .

$\frac{- 1 (\ln (7) - \ln (9))}{\ln (147)}$

Step 3.4.2

Move the negative in front of the fraction.

$- \frac{\ln (7) - \ln (9)}{\ln (147)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{\ln (7) - \ln (9)}{\ln (147)}$

Decimal Form:

$0.05035924 \dots$