Precalculus Examples

$\frac{2 \log (12) - 3 \log (25)}{\ln (4) + 3 \ln (6)}$

Step 1

Simplify the numerator.

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Step 1.1

Simplify $2 \log (12)$ by moving $2$ inside the logarithm.

$\frac{\log (12^{2}) - 3 \log (25)}{\ln (4) + 3 \ln (6)}$

Step 1.2

Raise $12$ to the power of $2$ .

$\frac{\log (144) - 3 \log (25)}{\ln (4) + 3 \ln (6)}$

Step 1.3

Simplify $- 3 \log (25)$ by moving $3$ inside the logarithm.

$\frac{\log (144) - \log (25^{3})}{\ln (4) + 3 \ln (6)}$

Step 1.4

Raise $25$ to the power of $3$ .

$\frac{\log (144) - \log (15625)}{\ln (4) + 3 \ln (6)}$

Step 1.5

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

$\frac{\log (\frac{144}{15625})}{\ln (4) + 3 \ln (6)}$

Step 2

Simplify the denominator.

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Step 2.1

Simplify $3 \ln (6)$ by moving $3$ inside the logarithm.

$\frac{\log (\frac{144}{15625})}{\ln (4) + \ln (6^{3})}$

Step 2.2

Raise $6$ to the power of $3$ .

$\frac{\log (\frac{144}{15625})}{\ln (4) + \ln (216)}$

Step 2.3

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

$\frac{\log (\frac{144}{15625})}{\ln (4 \cdot 216)}$

Step 2.4

Multiply $4$ by $216$ .

$\frac{\log (\frac{144}{15625})}{\ln (864)}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\frac{144}{15625})}{\ln (864)}$

Decimal Form:

$- 0.30103314 \dots$