Precalculus Examples

$\frac{- 3 \log (5) - \log (3)}{2 \log (5) - \log (3)}$

Step 1

Simplify the numerator.

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Step 1.1

Simplify $- 3 \log (5)$ by moving $3$ inside the logarithm.

$\frac{- \log (5^{3}) - \log (3)}{2 \log (5) - \log (3)}$

Step 1.2

Raise $5$ to the power of $3$ .

$\frac{- \log (125) - \log (3)}{2 \log (5) - \log (3)}$

Step 2

Simplify the denominator.

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Step 2.1

Simplify $2 \log (5)$ by moving $2$ inside the logarithm.

$\frac{- \log (125) - \log (3)}{\log (5^{2}) - \log (3)}$

Step 2.2

Raise $5$ to the power of $2$ .

$\frac{- \log (125) - \log (3)}{\log (25) - \log (3)}$

Step 2.3

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

$\frac{- \log (125) - \log (3)}{\log (\frac{25}{3})}$

Step 3

Simplify with factoring out.

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Step 3.1

Factor $- 1$ out of $- \log (125)$ .

$\frac{- (\log (125)) - \log (3)}{\log (\frac{25}{3})}$

Step 3.2

Factor $- 1$ out of $- \log (3)$ .

$\frac{- (\log (125)) - (\log (3))}{\log (\frac{25}{3})}$

Step 3.3

Factor $- 1$ out of $- (\log (125)) - (\log (3))$ .

$\frac{- (\log (125) + \log (3))}{\log (\frac{25}{3})}$

Step 3.4

Simplify the expression.

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Step 3.4.1

Rewrite $- (\log (125) + \log (3))$ as $- 1 (\log (125) + \log (3))$ .

$\frac{- 1 (\log (125) + \log (3))}{\log (\frac{25}{3})}$

Step 3.4.2

Move the negative in front of the fraction.

$- \frac{\log (125) + \log (3)}{\log (\frac{25}{3})}$

Step 4

Simplify the numerator.

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Step 4.1

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

$- \frac{\log (125 \cdot 3)}{\log (\frac{25}{3})}$

Step 4.2

Multiply $125$ by $3$ .

$- \frac{\log (375)}{\log (\frac{25}{3})}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{\log (375)}{\log (\frac{25}{3})}$

Decimal Form:

$- 2.79537233 \dots$