Precalculus Examples

$\frac{\ln (2197)}{- \ln (13) + 2 \ln (14)}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $\ln (2197)$ as $\ln (13^{3})$ .

$\frac{\ln (13^{3})}{- \ln (13) + 2 \ln (14)}$

Step 1.2

Expand $\ln (13^{3})$ by moving $3$ outside the logarithm.

$\frac{3 \ln (13)}{- \ln (13) + 2 \ln (14)}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $\ln (14)$ as $\ln (2 (7))$ .

$\frac{3 \ln (13)}{- \ln (13) + 2 \ln (2 (7))}$

Step 2.2

Rewrite $\ln (2 (7))$ as $\ln (2) + \ln (7)$ .

$\frac{3 \ln (13)}{- \ln (13) + 2 (\ln (2) + \ln (7))}$

Step 2.3

Apply the distributive property.

$\frac{3 \ln (13)}{- \ln (13) + 2 \ln (2) + 2 \ln (7)}$

Step 3

Factor $- 1$ out of $- \ln (13)$ .

$\frac{3 \ln (13)}{- (\ln (13)) + 2 \ln (2) + 2 \ln (7)}$

Step 4

Factor $- 1$ out of $2 \ln (2)$ .

$\frac{3 \ln (13)}{- (\ln (13)) - (- 2 \ln (2)) + 2 \ln (7)}$

Step 5

Factor $- 1$ out of $- (\ln (13)) - (- 2 \ln (2))$ .

$\frac{3 \ln (13)}{- (\ln (13) - 2 \ln (2)) + 2 \ln (7)}$

Step 6

Factor $- 1$ out of $2 \ln (7)$ .

$\frac{3 \ln (13)}{- (\ln (13) - 2 \ln (2)) - (- 2 \ln (7))}$

Step 7

Factor $- 1$ out of $- (\ln (13) - 2 \ln (2)) - (- 2 \ln (7))$ .

$\frac{3 \ln (13)}{- (\ln (13) - 2 \ln (2) - 2 \ln (7))}$

Step 8

Rewrite $- (\ln (13) - 2 \ln (2) - 2 \ln (7))$ as $- 1 (\ln (13) - 2 \ln (2) - 2 \ln (7))$ .

$\frac{3 \ln (13)}{- 1 (\ln (13) - 2 \ln (2) - 2 \ln (7))}$

Step 9

Move the negative in front of the fraction.

$- \frac{3 \ln (13)}{\ln (13) - 2 \ln (2) - 2 \ln (7)}$