Precalculus Examples

$\frac{3 \ln (7) - 3 \ln (3)}{4 \ln (3) - \ln (7)}$

Step 1

Simplify the numerator.

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Step 1.1

Simplify $3 \ln (7)$ by moving $3$ inside the logarithm.

$\frac{\ln (7^{3}) - 3 \ln (3)}{4 \ln (3) - \ln (7)}$

Step 1.2

Raise $7$ to the power of $3$ .

$\frac{\ln (343) - 3 \ln (3)}{4 \ln (3) - \ln (7)}$

Step 1.3

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

$\frac{\ln (343) - \ln (3^{3})}{4 \ln (3) - \ln (7)}$

Step 1.4

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

$\frac{\ln (343) - \ln (27)}{4 \ln (3) - \ln (7)}$

Step 1.5

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

$\frac{\ln (\frac{343}{27})}{4 \ln (3) - \ln (7)}$

Step 2

Simplify the denominator.

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Step 2.1

Simplify $4 \ln (3)$ by moving $4$ inside the logarithm.

$\frac{\ln (\frac{343}{27})}{\ln (3^{4}) - \ln (7)}$

Step 2.2

Raise $3$ to the power of $4$ .

$\frac{\ln (\frac{343}{27})}{\ln (81) - \ln (7)}$

Step 2.3

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

$\frac{\ln (\frac{343}{27})}{\ln (\frac{81}{7})}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (\frac{343}{27})}{\ln (\frac{81}{7})}$

Decimal Form:

$1.03812664 \dots$